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Chapter 3: Graphing

## 3.4 Graphing Linear Equations

There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation.

If the equation is given in the form [latex]y = mx + b[/latex], then [latex]m[/latex] gives the rise over run value and the value [latex]b[/latex] gives the point where the line crosses the [latex]y[/latex]-axis, also known as the [latex]y[/latex]-intercept.

Example 3.4.1

Given the following equations, identify the slope and the [latex]y[/latex]-intercept.

- [latex]\begin{array}{lll} y = 2x - 3\hspace{0.14in} & \text{Slope }(m)=2\hspace{0.1in}&y\text{-intercept } (b)=-3 \end{array}[/latex]
- [latex]\begin{array}{lll} y = \dfrac{1}{2}x - 1\hspace{0.08in} & \text{Slope }(m)=\dfrac{1}{2}\hspace{0.1in}&y\text{-intercept } (b)=-1 \end{array}[/latex]
- [latex]\begin{array}{lll} y = -3x + 4 & \text{Slope }(m)=-3 &y\text{-intercept } (b)=4 \end{array}[/latex]
- [latex]\begin{array}{lll} y = \dfrac{2}{3}x\hspace{0.34in} & \text{Slope }(m)=\dfrac{2}{3}\hspace{0.1in} &y\text{-intercept } (b)=0 \end{array}[/latex]

When graphing a linear equation using the slope-intercept method, start by using the value given for the [latex]y[/latex]-intercept. After this point is marked, then identify other points using the slope.

This is shown in the following example.

Example 3.4.2

Graph the equation [latex]y = 2x - 3[/latex].

First, place a dot on the [latex]y[/latex]-intercept, [latex]y = -3[/latex], which is placed on the coordinate [latex](0, -3).[/latex]

Now, place the next dot using the slope of 2.

A slope of 2 means that the line rises 2 for every 1 across.

Simply, [latex]m = 2[/latex] is the same as [latex]m = \dfrac{2}{1}[/latex], where [latex]\Delta y = 2[/latex] and [latex]\Delta x = 1[/latex].

Placing these points on the graph becomes a simple counting exercise, which is done as follows:

Once several dots have been drawn, draw a line through them, like so:

Note that dots can also be drawn in the reverse of what has been drawn here.

Slope is 2 when rise over run is [latex]\dfrac{2}{1}[/latex] or [latex]\dfrac{-2}{-1}[/latex], which would be drawn as follows:

Example 3.4.3

Graph the equation [latex]y = \dfrac{2}{3}x[/latex].

First, place a dot on the [latex]y[/latex]-intercept, [latex](0, 0)[/latex].

Now, place the dots according to the slope, [latex]\dfrac{2}{3}[/latex].

This will generate the following set of dots on the graph. All that remains is to draw a line through the dots.

The second method of drawing lines represented by linear equations and functions is to identify the two intercepts of the linear equation. Specifically, find [latex]x[/latex] when [latex]y = 0[/latex] and find [latex]y[/latex] when [latex]x = 0[/latex].

Example 3.4.4

Graph the equation [latex]2x + y = 6[/latex].

To find the first coordinate, choose [latex]x = 0[/latex].

This yields:

[latex]\begin{array}{lllll} 2(0)&+&y&=&6 \\ &&y&=&6 \end{array}[/latex]

Coordinate is [latex](0, 6)[/latex].

Now choose [latex]y = 0[/latex].

[latex]\begin{array}{llrll} 2x&+&0&=&6 \\ &&2x&=&6 \\ &&x&=&\frac{6}{2} \text{ or } 3 \end{array}[/latex]

Coordinate is [latex](3, 0)[/latex].

Draw these coordinates on the graph and draw a line through them.

Example 3.4.5

Graph the equation [latex]x + 2y = 4[/latex].

[latex]\begin{array}{llrll} (0)&+&2y&=&4 \\ &&y&=&\frac{4}{2} \text{ or } 2 \end{array}[/latex]

Coordinate is [latex](0, 2)[/latex].

[latex]\begin{array}{llrll} x&+&2(0)&=&4 \\ &&x&=&4 \end{array}[/latex]

Coordinate is [latex](4, 0)[/latex].

Example 3.4.6

Graph the equation [latex]2x + y = 0[/latex].

[latex]\begin{array}{llrll} 2(0)&+&y&=&0 \\ &&y&=&0 \end{array}[/latex]

Coordinate is [latex](0, 0)[/latex].

Since the intercept is [latex](0, 0)[/latex], finding the other intercept yields the same coordinate. In this case, choose any value of convenience.

Choose [latex]x = 2[/latex].

[latex]\begin{array}{rlrlr} 2(2)&+&y&=&0 \\ 4&+&y&=&0 \\ -4&&&&-4 \\ \hline &&y&=&-4 \end{array}[/latex]

Coordinate is [latex](2, -4)[/latex].

For questions 1 to 10, sketch each linear equation using the slope-intercept method.

- [latex]y = -\dfrac{1}{4}x - 3[/latex]
- [latex]y = \dfrac{3}{2}x - 1[/latex]
- [latex]y = -\dfrac{5}{4}x - 4[/latex]
- [latex]y = -\dfrac{3}{5}x + 1[/latex]
- [latex]y = -\dfrac{4}{3}x + 2[/latex]
- [latex]y = \dfrac{5}{3}x + 4[/latex]
- [latex]y = \dfrac{3}{2}x - 5[/latex]
- [latex]y = -\dfrac{2}{3}x - 2[/latex]
- [latex]y = -\dfrac{4}{5}x - 3[/latex]
- [latex]y = \dfrac{1}{2}x[/latex]

For questions 11 to 20, sketch each linear equation using the [latex]x\text{-}[/latex] and [latex]y[/latex]-intercepts.

- [latex]x + 4y = -4[/latex]
- [latex]2x - y = 2[/latex]
- [latex]2x + y = 4[/latex]
- [latex]3x + 4y = 12[/latex]
- [latex]4x + 3y = -12[/latex]
- [latex]x + y = -5[/latex]
- [latex]3x + 2y = 6[/latex]
- [latex]x - y = -2[/latex]
- [latex]4x - y = -4[/latex]

For questions 21 to 28, sketch each linear equation using any method.

- [latex]y = -\dfrac{1}{2}x + 3[/latex]
- [latex]y = 2x - 1[/latex]
- [latex]y = -\dfrac{5}{4}x[/latex]
- [latex]y = -3x + 2[/latex]
- [latex]y = -\dfrac{3}{2}x + 1[/latex]
- [latex]y = \dfrac{1}{3}x - 3[/latex]
- [latex]y = \dfrac{3}{2}x + 2[/latex]
- [latex]y = 2x - 2[/latex]

For questions 29 to 40, reduce and sketch each linear equation using any method.

- [latex]y + 3 = -\dfrac{4}{5}x + 3[/latex]
- [latex]y - 4 = \dfrac{1}{2}x[/latex]
- [latex]x + 5y = -3 + 2y[/latex]
- [latex]3x - y = 4 + x - 2y[/latex]
- [latex]4x + 3y = 5 (x + y)[/latex]
- [latex]3x + 4y = 12 - 2y[/latex]
- [latex]2x - y = 2 - y \text{ (tricky)}[/latex]
- [latex]7x + 3y = 2(2x + 2y) + 6[/latex]
- [latex]x + y = -2x + 3[/latex]
- [latex]3x + 4y = 3y + 6[/latex]
- [latex]2(x + y) = -3(x + y) + 5[/latex]
- [latex]9x - y = 4x + 5[/latex]

Answer Key 3.4

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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## Unit 4: Linear equations and linear systems

Lesson 3: balanced moves.

- Intro to equations with variables on both sides (Opens a modal)
- Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
- Equations with variables on both sides Get 3 of 4 questions to level up!

## Lesson 4: More balanced moves

- Equations with parentheses (Opens a modal)
- Equations with parentheses Get 3 of 4 questions to level up!

## Lesson 5: Solving any linear equation

- Multi-step equations review (Opens a modal)

## Lesson 6: Strategic solving

- No videos or articles available in this lesson
- Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!
- Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up!

## Extra practice: Linear equations

- Sums of consecutive integers (Opens a modal)
- Sum of integers challenge (Opens a modal)
- Equation practice with vertical angles (Opens a modal)
- Equation practice with complementary angles (Opens a modal)
- Equation practice with supplementary angles (Opens a modal)
- Sums of consecutive integers Get 3 of 4 questions to level up!
- Equation practice with vertical angles Get 3 of 4 questions to level up!
- Equation practice with angle addition Get 3 of 4 questions to level up!

## Lesson 7: All, some, or no solutions

- Creating an equation with no solutions (Opens a modal)
- Creating an equation with infinitely many solutions (Opens a modal)
- Number of solutions to equations challenge Get 3 of 4 questions to level up!

## Lesson 8: How many solutions?

- Number of solutions to equations (Opens a modal)
- Worked example: number of solutions to equations (Opens a modal)
- Number of solutions to equations Get 3 of 4 questions to level up!

## Lesson 9: When are they the same?

- Age word problem: Imran (Opens a modal)
- Age word problem: Ben & William (Opens a modal)
- Age word problem: Arman & Diya (Opens a modal)
- Age word problems Get 3 of 4 questions to level up!

## Lesson 10: On or off the line?

- Solutions of systems of equations Get 3 of 4 questions to level up!

## Lesson 12: Systems of equations

- Systems of equations: trolls, tolls (1 of 2) (Opens a modal)
- Systems of equations: trolls, tolls (2 of 2) (Opens a modal)
- Systems of equations with graphing (Opens a modal)
- Systems of equations with graphing: y=7/5x-5 & y=3/5x-1 (Opens a modal)
- Systems of equations with graphing Get 3 of 4 questions to level up!
- Number of solutions to a system of equations graphically Get 3 of 4 questions to level up!

## Lesson 13: Solving systems of equations

- Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120 (Opens a modal)
- Number of solutions to a system of equations graphically (Opens a modal)
- Number of solutions to system of equations review (Opens a modal)
- Number of solutions to a system of equations algebraically Get 3 of 4 questions to level up!

## Lesson 14: Solving more systems

- Systems of equations with substitution: 2y=x+7 & x=y-4 (Opens a modal)
- Systems of equations with substitution (Opens a modal)
- Systems of equations with substitution: y=4x-17.5 & y+2x=6.5 (Opens a modal)
- Systems of equations with substitution: y=-5x+8 & 10x+2y=-2 (Opens a modal)
- Substitution method review (systems of equations) (Opens a modal)
- Systems of equations with substitution Get 3 of 4 questions to level up!

## Lesson 16: Solving problems with systems of equations

- System of equations word problem: no solution (Opens a modal)
- Systems of equations with substitution: coins (Opens a modal)
- Systems of equations word problems Get 3 of 4 questions to level up!

## 3.1 Graph Linear Equations in Two Variables

Learning objectives.

By the end of this section, you will be able to:

- Plot points in a rectangular coordinate system
- Graph a linear equation by plotting points
- Graph vertical and horizontal lines
- Find the x- and y-intercepts
- Graph a line using the intercepts

## Be Prepared 3.1

Before you get started, take this readiness quiz.

Evaluate 5 x − 4 5 x − 4 when x = −1 . x = −1 . If you missed this problem, review Example 1.6 .

## Be Prepared 3.2

Evaluate 3 x − 2 y 3 x − 2 y when x = 4 , y = −3 . x = 4 , y = −3 . If you missed this problem, review Example 1.21 .

## Be Prepared 3.3

Solve for y : 8 − 3 y = 20 . 8 − 3 y = 20 . If you missed this problem, review Example 2.2 .

## Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy -plane or the “coordinate plane.”

The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the x -axis. The vertical number line is called the y -axis. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See Figure 3.2 .

In the rectangular coordinate system, every point is represented by an ordered pair . The first number in the ordered pair is the x -coordinate of the point, and the second number is the y -coordinate of the point. The phrase “ordered pair” means that the order is important.

## Ordered Pair

An ordered pair , ( x , y ) ( x , y ) gives the coordinates of a point in a rectangular coordinate system. The first number is the x -coordinate. The second number is the y -coordinate.

What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is ( 0 , 0 ) . ( 0 , 0 ) . The point ( 0 , 0 ) ( 0 , 0 ) has a special name. It is called the origin .

The point ( 0 , 0 ) ( 0 , 0 ) is called the origin . It is the point where the x -axis and y -axis intersect.

We use the coordinates to locate a point on the xy -plane. Let’s plot the point ( 1 , 3 ) ( 1 , 3 ) as an example. First, locate 1 on the x -axis and lightly sketch a vertical line through x = 1 . x = 1 . Then, locate 3 on the y -axis and sketch a horizontal line through y = 3 . y = 3 . Now, find the point where these two lines meet—that is the point with coordinates ( 1 , 3 ) . ( 1 , 3 ) . See Figure 3.3 .

Notice that the vertical line through x = 1 x = 1 and the horizontal line through y = 3 y = 3 are not part of the graph. We just used them to help us locate the point ( 1 , 3 ) . ( 1 , 3 ) .

When one of the coordinate is zero, the point lies on one of the axes. In Figure 3.4 the point ( 0 , 4 ) ( 0 , 4 ) is on the y -axis and the point ( −2 , 0 ) ( −2 , 0 ) is on the x -axis.

## Points on the Axes

Points with a y -coordinate equal to 0 are on the x -axis, and have coordinates ( a , 0 ) . ( a , 0 ) .

Points with an x -coordinate equal to 0 are on the y -axis, and have coordinates ( 0 , b ) . ( 0 , b ) .

## Example 3.1

Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:

ⓐ ( −5 , 4 ) ( −5 , 4 ) ⓑ ( −3 , −4 ) ( −3 , −4 ) ⓒ ( 2 , −3 ) ( 2 , −3 ) ⓓ ( 0 , −1 ) ( 0 , −1 ) ⓔ ( 3 , 5 2 ) . ( 3 , 5 2 ) .

The first number of the coordinate pair is the x -coordinate, and the second number is the y -coordinate. To plot each point, sketch a vertical line through the x -coordinate and a horizontal line through the y -coordinate. Their intersection is the point. ⓐ Since x = −5 , x = −5 , the point is to the left of the y -axis. Also, since y = 4 , y = 4 , the point is above the x -axis. The point ( −5 , 4 ) ( −5 , 4 ) is in Quadrant II. ⓑ Since x = −3 , x = −3 , the point is to the left of the y -axis. Also, since y = −4 , y = −4 , the point is below the x -axis. The point ( −3 , −4 ) ( −3 , −4 ) is in Quadrant III. ⓒ Since x = 2 , x = 2 , the point is to the right of the y -axis. Since y = −3 , y = −3 , the point is below the x -axis. The point ( 2 , −3 ) ( 2 , −3 ) is in Quadrant IV. ⓓ Since x = 0 , x = 0 , the point whose coordinates are ( 0 , −1 ) ( 0 , −1 ) is on the y -axis. ⓔ Since x = 3 , x = 3 , the point is to the right of the y -axis. Since y = 5 2 , y = 5 2 , the point is above the x -axis. (It may be helpful to write 5 2 5 2 as a mixed number or decimal.) The point ( 3 , 5 2 ) ( 3 , 5 2 ) is in Quadrant I.

Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located: ⓐ ( −2 , 1 ) ( −2 , 1 ) ⓑ ( −3 , −1 ) ( −3 , −1 ) ⓒ ( 4 , −4 ) ( 4 , −4 ) ⓓ ( −4 , 4 ) ( −4 , 4 ) ⓔ ( −4 , 3 2 ) ( −4 , 3 2 )

Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located: ⓐ ( −4 , 1 ) ( −4 , 1 ) ⓑ ( −2 , 3 ) ( −2 , 3 ) ⓒ ( 2 , −5 ) ( 2 , −5 ) ⓓ ( −2 , 5 ) ( −2 , 5 ) ⓔ ( −3 , 5 2 ) ( −3 , 5 2 )

The signs of the x -coordinate and y -coordinate affect the location of the points. You may have noticed some patterns as you graphed the points in the previous example. We can summarize sign patterns of the quadrants in this way:

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. But equations can have more than one variable. Equations with two variables may be of the form A x + B y = C . A x + B y = C . An equation of this form is called a linear equation in two variables.

## Linear Equation

An equation of the form A x + B y = C , A x + B y = C , where A and B are not both zero, is called a linear equation in two variables.

Here is an example of a linear equation in two variables, x and y .

The equation y = −3 x + 5 y = −3 x + 5 is also a linear equation. But it does not appear to be in the form A x + B y = C . A x + B y = C . We can use the Addition Property of Equality and rewrite it in A x + B y = C A x + B y = C form.

By rewriting y = −3 x + 5 y = −3 x + 5 as 3 x + y = 5 , 3 x + y = 5 , we can easily see that it is a linear equation in two variables because it is of the form A x + B y = C . A x + B y = C . When an equation is in the form A x + B y = C , A x + B y = C , we say it is in standard form of a linear equation .

## Standard Form of Linear Equation

A linear equation is in standard form when it is written A x + B y = C . A x + B y = C .

Most people prefer to have A , B , and C be integers and A ≥ 0 A ≥ 0 when writing a linear equation in standard form, although it is not strictly necessary.

Linear equations have infinitely many solutions. For every number that is substituted for x there is a corresponding y value. This pair of values is a solution to the linear equation and is represented by the ordered pair ( x , y ) . ( x , y ) . When we substitute these values of x and y into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.

## Solution of a Linear Equation in Two Variables

An ordered pair ( x , y ) ( x , y ) is a solution of the linear equation A x + B y = C , A x + B y = C , if the equation is a true statement when the x - and y -values of the ordered pair are substituted into the equation.

Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.

A graph is a visual representation of all the solutions of the equation. It is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to that equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation. Points not on the line are not solutions!

## Graph of a Linear Equation

The graph of a linear equation A x + B y = C A x + B y = C is a straight line.

- Every point on the line is a solution of the equation.
- Every solution of this equation is a point on this line.

## Example 3.2

The graph of y = 2 x − 3 y = 2 x − 3 is shown.

For each ordered pair, decide:

ⓐ Is the ordered pair a solution to the equation?

ⓑ Is the point on the line?

A: ( 0 , −3 ) ( 0 , −3 ) B: ( 3 , 3 ) ( 3 , 3 ) C: ( 2 , −3 ) ( 2 , −3 ) D: ( −1 , −5 ) ( −1 , −5 )

Substitute the x - and y -values into the equation to check if the ordered pair is a solution to the equation. ⓐ

ⓑ Plot the points ( 0 , −3 ) , ( 0 , −3 ) , ( 3 , 3 ) , ( 3 , 3 ) , ( 2 , −3 ) , ( 2 , −3 ) , and ( −1 , −5 ) . ( −1 , −5 ) .

The points ( 0 , 3 ) , ( 0 , 3 ) , ( 3 , −3 ) , ( 3 , −3 ) , and ( −1 , −5 ) ( −1 , −5 ) are on the line y = 2 x − 3 , y = 2 x − 3 , and the point ( 2 , −3 ) ( 2 , −3 ) is not on the line. The points that are solutions to y = 2 x − 3 y = 2 x − 3 are on the line, but the point that is not a solution is not on the line.

Use graph of y = 3 x − 1 . y = 3 x − 1 . For each ordered pair, decide:

ⓐ Is the ordered pair a solution to the equation? ⓑ Is the point on the line?

A ( 0 , −1 ) ( 0 , −1 ) B ( 2 , 5 ) ( 2 , 5 )

A ( 3 , −1 ) ( 3 , −1 ) B ( −1 , −4 ) ( −1 , −4 )

Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.

## Example 3.3

How to graph a linear equation by plotting points.

Graph the equation y = 2 x + 1 y = 2 x + 1 by plotting points.

Graph the equation by plotting points: y = 2 x − 3 . y = 2 x − 3 .

Graph the equation by plotting points: y = −2 x + 4 . y = −2 x + 4 .

The steps to take when graphing a linear equation by plotting points are summarized here.

## Graph a linear equation by plotting points.

- Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.

If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between these illustrations.

When an equation includes a fraction as the coefficient of x x , we can still substitute any numbers for x . But the arithmetic is easier if we make “good” choices for the values of x . This way we will avoid fractional answers, which are hard to graph precisely.

## Example 3.4

Graph the equation: y = 1 2 x + 3 . y = 1 2 x + 3 .

Find three points that are solutions to the equation. Since this equation has the fraction 1 2 1 2 as a coefficient of x , we will choose values of x carefully. We will use zero as one choice and multiples of 2 for the other choices. Why are multiples of two a good choice for values of x ? By choosing multiples of 2 the multiplication by 1 2 1 2 simplifies to a whole number

The points are shown in Table 3.1 .

0 | 3 | |

2 | 4 | |

4 | 5 |

Plot the points, check that they line up, and draw the line.

Graph the equation: y = 1 3 x − 1 . y = 1 3 x − 1 .

Graph the equation: y = 1 4 x + 2 . y = 1 4 x + 2 .

## Graph Vertical and Horizontal Lines

Some linear equations have only one variable. They may have just x and no y , or just y without an x . This changes how we make a table of values to get the points to plot.

Let’s consider the equation x = −3 . x = −3 . This equation has only one variable, x . The equation says that x is always equal to −3 , −3 , so its value does not depend on y . No matter what is the value of y , the value of x is always −3 . −3 .

So to make a table of values, write −3 −3 in for all the x -values. Then choose any values for y . Since x does not depend on y , you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y -coordinates. See Table 3.2 .

1 | ||

2 | ||

3 |

Plot the points from the table and connect them with a straight line. Notice that we have graphed a vertical line .

What if the equation has y but no x ? Let’s graph the equation y = 4 . y = 4 . This time the y- value is a constant, so in this equation, y does not depend on x . Fill in 4 for all the y ’s in Table 3.3 and then choose any values for x . We’ll use 0, 2, and 4 for the x -coordinates.

0 | 4 | |

2 | 4 | |

4 | 4 |

In this figure, we have graphed a horizontal line passing through the y -axis at 4.

## Vertical and Horizontal Lines

A vertical line is the graph of an equation of the form x = a . x = a .

The line passes through the x -axis at ( a , 0 ) . ( a , 0 ) .

A horizontal line is the graph of an equation of the form y = b . y = b .

The line passes through the y -axis at ( 0 , b ) . ( 0 , b ) .

## Example 3.5

Graph: ⓐ x = 2 x = 2 ⓑ y = −1 . y = −1 .

ⓐ The equation has only one variable, x , and x is always equal to 2. We create a table where x is always 2 and then put in any values for y . The graph is a vertical line passing through the x -axis at 2.

2 | 1 | |

2 | 2 | |

2 | 3 |

ⓑ Similarly, the equation y = −1 y = −1 has only one variable, y . The value of y is constant. All the ordered pairs in the next table have the same y -coordinate. The graph is a horizontal line passing through the y -axis at −1 . −1 .

0 | ||

3 | ||

Graph the equations: ⓐ x = 5 x = 5 ⓑ y = −4 . y = −4 .

## Try It 3.10

Graph the equations: ⓐ x = −2 x = −2 ⓑ y = 3 . y = 3 .

What is the difference between the equations y = 4 x y = 4 x and y = 4 ? y = 4 ?

The equation y = 4 x y = 4 x has both x and y . The value of y depends on the value of x , so the y -coordinate changes according to the value of x . The equation y = 4 y = 4 has only one variable. The value of y is constant, it does not depend on the value of x , so the y -coordinate is always 4.

Notice, in the graph, the equation y = 4 x y = 4 x gives a slanted line, while y = 4 y = 4 gives a horizontal line.

## Example 3.6

Graph y = −3 x y = −3 x and y = −3 y = −3 in the same rectangular coordinate system.

We notice that the first equation has the variable x , while the second does not. We make a table of points for each equation and then graph the lines. The two graphs are shown.

## Try It 3.11

Graph the equations in the same rectangular coordinate system: y = −4 x y = −4 x and y = −4 . y = −4 .

## Try It 3.12

Graph the equations in the same rectangular coordinate system: y = 3 y = 3 and y = 3 x . y = 3 x .

## Find x - and y -intercepts

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x -axis and the y -axis. These points are called the intercepts of a line .

## Intercepts of a Line

The points where a line crosses the x -axis and the y -axis are called the intercepts of the line .

Let’s look at the graphs of the lines.

First, notice where each of these lines crosses the x -axis. See Table 3.4 .

Now, let’s look at the points where these lines cross the y -axis.

the -axis at: | for this point | the axis at: | for this point | |
---|---|---|---|---|

Figure (a) | 3 | 6 | ||

Figure (b) | 4 | |||

Figure (c) | 5 | |||

Figure (d) | 0 | 0 | ||

General Figure |

Do you see a pattern?

For each line, the y -coordinate of the point where the line crosses the x -axis is zero. The point where the line crosses the x -axis has the form ( a , 0 ) ( a , 0 ) and is called the x-intercept of the line. The x -intercept occurs when y is zero.

In each line, the x - coordinate of the point where the line crosses the y -axis is zero. The point where the line crosses the y -axis has the form ( 0 , b ) ( 0 , b ) and is called the y-intercept of the line. The y -intercept occurs when x is zero.

## x -intercept and y -intercept of a Line

The x -intercept is the point ( a , 0 ) ( a , 0 ) where the line crosses the x -axis.

The y -intercept is the point ( 0 , b ) ( 0 , b ) where the line crosses the y -axis.

## Example 3.7

Find the x - and y -intercepts on each graph shown.

ⓐ The graph crosses the x -axis at the point ( 4 , 0 ) . ( 4 , 0 ) . The x- intercept is ( 4 , 0 ) . ( 4 , 0 ) . The graph crosses the y -axis at the point ( 0 , 2 ) . ( 0 , 2 ) . The y -intercept is ( 0 , 2 ) . ( 0 , 2 ) . ⓑ The graph crosses the x -axis at the point ( 2 , 0 ) . ( 2 , 0 ) . The x -intercept is ( 2 , 0 ) . ( 2 , 0 ) . The graph crosses the y -axis at the point ( 0 , −6 ) . ( 0 , −6 ) . The y -intercept is ( 0 , −6 ) . ( 0 , −6 ) . ⓒ The graph crosses the x -axis at the point ( −5 , 0 ) . ( −5 , 0 ) . The x -intercept is ( −5 , 0 ) . ( −5 , 0 ) . The graph crosses the y -axis at the point ( 0 , −5 ) . ( 0 , −5 ) . The y -intercept is ( 0 , −5 ) . ( 0 , −5 ) .

## Try It 3.13

Find the x - and y -intercepts on the graph.

## Try It 3.14

Recognizing that the x -intercept occurs when y is zero and that the y -intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x -intercept, let y = 0 y = 0 and solve for x . To find the y -intercept, let x = 0 x = 0 and solve for y .

## Find the x - and y -intercepts from the Equation of a Line

Use the equation of the line. To find:

- the x -intercept of the line, let y = 0 y = 0 and solve for x .
- the y -intercept of the line, let x = 0 x = 0 and solve for y .

## Example 3.8

Find the intercepts of 2 x + y = 8 . 2 x + y = 8 .

We will let y = 0 y = 0 to find the x -intercept, and let x = 0 x = 0 to find the y -intercept. We will fill in a table, which reminds us of what we need to find.

To find the -intercept, let | |

Let | |

Simplify. | |

The -intercept is: | |

To find the -intercept, let | |

Let | |

Simplify. | |

The -intercept is: |

The intercepts are the points ( 4 , 0 ) ( 4 , 0 ) and ( 0 , 8 ) ( 0 , 8 ) as shown in the table.

4 | 0 |

0 | 8 |

## Try It 3.15

Find the intercepts: 3 x + y = 12 . 3 x + y = 12 .

## Try It 3.16

Find the intercepts: x + 4 y = 8 . x + 4 y = 8 .

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

## Example 3.9

How to graph a line using the intercepts.

Graph – x + 2 y = 6 – x + 2 y = 6 using the intercepts.

## Try It 3.17

Graph using the intercepts: x – 2 y = 4 . x – 2 y = 4 .

## Try It 3.18

Graph using the intercepts: – x + 3 y = 6 . – x + 3 y = 6 .

The steps to graph a linear equation using the intercepts are summarized here.

## Graph a linear equation using the intercepts.

- Let y = 0 y = 0 and solve for x .
- Let x = 0 x = 0 and solve for y .
- Step 2. Find a third solution to the equation.
- Step 3. Plot the three points and check that they line up.
- Step 4. Draw the line.

## Example 3.10

Graph 4 x − 3 y = 12 4 x − 3 y = 12 using the intercepts.

Find the intercepts and a third point.

We list the points in the table and show the graph.

3 | 0 | |

0 | ||

6 | 4 |

## Try It 3.19

Graph using the intercepts: 5 x − 2 y = 10 . 5 x − 2 y = 10 .

## Try It 3.20

Graph using the intercepts: 3 x − 4 y = 12 . 3 x − 4 y = 12 .

When the line passes through the origin, the x -intercept and the y -intercept are the same point.

## Example 3.11

Graph y = 5 x y = 5 x using the intercepts.

This line has only one intercept. It is the point ( 0 , 0 ) . ( 0 , 0 ) . To ensure accuracy, we need to plot three points. Since the x - and y -intercepts are the same point, we need two more points to graph the line.

The resulting three points are summarized in the table.

0 | 0 | |

1 | 5 | |

Plot the three points, check that they line up, and draw the line.

## Try It 3.21

Graph using the intercepts: y = 4 x . y = 4 x .

## Try It 3.22

Graph the intercepts: y = − x . y = − x .

## Section 3.1 Exercises

Practice makes perfect.

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

ⓐ ( −4 , 2 ) ( −4 , 2 ) ⓑ ( −1 , −2 ) ( −1 , −2 ) ⓒ ( 3 , −5 ) ( 3 , −5 ) ⓓ ( −3 , 0 ) ( −3 , 0 ) ⓔ ( 5 3 , 2 ) ( 5 3 , 2 )

ⓐ ( −2 , −3 ) ( −2 , −3 ) ⓑ ( 3 , −3 ) ( 3 , −3 ) ⓒ ( −4 , 1 ) ( −4 , 1 ) ⓓ ( 4 , −1 ) ( 4 , −1 ) ⓔ ( 3 2 , 1 ) ( 3 2 , 1 )

ⓐ ( 3 , −1 ) ( 3 , −1 ) ⓑ ( −3 , 1 ) ( −3 , 1 ) ⓒ ( −2 , 0 ) ( −2 , 0 ) ⓓ ( −4 , −3 ) ( −4 , −3 ) ⓔ ( 1 , 14 5 ) ( 1 , 14 5 )

ⓐ ( −1 , 1 ) ( −1 , 1 ) ⓑ ( −2 , −1 ) ( −2 , −1 ) ⓒ ( 2 , 0 ) ( 2 , 0 ) ⓓ ( 1 , −4 ) ( 1 , −4 ) ⓔ ( 3 , 7 2 ) ( 3 , 7 2 )

In the following exercises, for each ordered pair, decide

ⓐ is the ordered pair a solution to the equation? ⓑ is the point on the line?

y = x + 2 ; y = x + 2 ; A: ( 0 , 2 ) ; ( 0 , 2 ) ; B: ( 1 , 2 ) ; ( 1 , 2 ) ; C: ( −1 , 1 ) ; ( −1 , 1 ) ; D: ( −3 , −1 ) . ( −3 , −1 ) .

y = x − 4 ; y = x − 4 ; A: ( 0 , −4 ) ; ( 0 , −4 ) ; B: ( 3 , −1 ) ; ( 3 , −1 ) ; C: ( 2 , 2 ) ; ( 2 , 2 ) ; D: ( 1 , −5 ) . ( 1 , −5 ) .

y = 1 2 x − 3 ; y = 1 2 x − 3 ; A: ( 0 , −3 ) ; ( 0 , −3 ) ; B: ( 2 , −2 ) ; ( 2 , −2 ) ; C: ( −2 , −4 ) ; ( −2 , −4 ) ; D: ( 4 , 1 ) ( 4 , 1 )

y = 1 3 x + 2 ; y = 1 3 x + 2 ; A: ( 0 , 2 ) ; ( 0 , 2 ) ; B: ( 3 , 3 ) ; ( 3 , 3 ) ; C: ( −3 , 2 ) ; ( −3 , 2 ) ; D: ( −6 , 0 ) . ( −6 , 0 ) .

In the following exercises, graph by plotting points.

y = x + 2 y = x + 2

y = x − 3 y = x − 3

y = 3 x − 1 y = 3 x − 1

y = −2 x + 2 y = −2 x + 2

y = − x − 3 y = − x − 3

y = − x − 2 y = − x − 2

y = 2 x y = 2 x

y = −2 x y = −2 x

y = 1 2 x + 2 y = 1 2 x + 2

y = 1 3 x − 1 y = 1 3 x − 1

y = 4 3 x − 5 y = 4 3 x − 5

y = 3 2 x − 3 y = 3 2 x − 3

y = − 2 5 x + 1 y = − 2 5 x + 1

y = − 4 5 x − 1 y = − 4 5 x − 1

y = − 3 2 x + 2 y = − 3 2 x + 2

y = − 5 3 x + 4 y = − 5 3 x + 4

Graph Vertical and Horizontal lines

In the following exercises, graph each equation.

ⓐ x = 4 x = 4 ⓑ y = 3 y = 3

ⓐ x = 3 x = 3 ⓑ y = 1 y = 1

ⓐ x = −2 x = −2 ⓑ y = −5 y = −5

ⓐ x = −5 x = −5 ⓑ y = −2 y = −2

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

y = 2 x y = 2 x and y = 2 y = 2

y = 5 x y = 5 x and y = 5 y = 5

y = − 1 2 x y = − 1 2 x and y = − 1 2 y = − 1 2

y = − 1 3 x y = − 1 3 x and y = − 1 3 y = − 1 3

Find x- and y- Intercepts

In the following exercises, find the x - and y -intercepts on each graph.

In the following exercises, find the intercepts for each equation.

x − y = 5 x − y = 5

x − y = −4 x − y = −4

3 x + y = 6 3 x + y = 6

x − 2 y = 8 x − 2 y = 8

4 x − y = 8 4 x − y = 8

5 x − y = 5 5 x − y = 5

2 x + 5 y = 10 2 x + 5 y = 10

3 x − 2 y = 12 3 x − 2 y = 12

In the following exercises, graph using the intercepts.

− x + 4 y = 8 − x + 4 y = 8

x + 2 y = 4 x + 2 y = 4

x + y = −3 x + y = −3

4 x + y = 4 4 x + y = 4

3 x + y = 3 3 x + y = 3

3 x − y = −6 3 x − y = −6

2 x − y = −8 2 x − y = −8

2 x + 4 y = 12 2 x + 4 y = 12

3 x − 2 y = 6 3 x − 2 y = 6

2 x − 5 y = −20 2 x − 5 y = −20

3 x − 4 y = −12 3 x − 4 y = −12

y = 5 x y = 5 x

y = x y = x

y = − x y = − x

Mixed Practice

y = 3 2 x y = 3 2 x

y = − 2 3 x y = − 2 3 x

y = − 1 2 x + 3 y = − 1 2 x + 3

y = 1 4 x − 2 y = 1 4 x − 2

4 x + y = 2 4 x + y = 2

5 x + 2 y = 10 5 x + 2 y = 10

y = −1 y = −1

x = 3 x = 3

## Writing Exercises

Explain how you would choose three x -values to make a table to graph the line y = 1 5 x − 2 . y = 1 5 x − 2 .

What is the difference between the equations of a vertical and a horizontal line?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation 4 x + y = −4 ? 4 x + y = −4 ? Why?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation y = 2 3 x − 2 ? y = 2 3 x − 2 ? Why?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

Confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

With some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

No, I don’t get it. This is a warning sign and you must address it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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## Unit 4 – Linear Functions and Arithmetic Sequences

This unit is all about understanding linear functions and using them to model real world scenarios. Fluency in interpreting the parameters of linear functions is emphasized as well as setting up linear functions to model a variety of situations. Linear inequalities are also taught. The unit ends with a introduction to sequences with an emphasis on arithmetic.

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Absolute Value and Step Functions

The Truth About Graphs

Graphs of Linear Inequalities

Introduction to Sequences

Arithmetic Sequences

Unit Review

Unit #4 Review – Linear Functions and Arithmetic Sequences

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EDITABLE ASSESSMENT

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Unit #4 Assessment Form C

Unit #4 Assessment Form D

Unit #4 Exit Tickets

Unit #4 Mid-Unit Quiz (Through Lesson #7) – Form A

Unit #4 Mid-Unit Quiz (Through Lesson #7) – Form B

Unit #4 Mid-Unit Quiz (Through Lesson #7) – Form C

U04.AO.01 – Writing the Equation of a Line Given Two Points.Extra Practice (After Lesson #6)

EDITABLE RESOURCE

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U04.AO.03 – Lesson #9.5 – Absolute Value Equations

U04.AO.04 – Turning Patterns into Sequences (after Lesson #13) – Enrichment Lesson

U04.AO.05 – Linear Modeling of Population (Extended Problem)

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## Linear Equations (Algebra 1 Curriculum - Unit 4) | All Things Algebra®

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3.4 Graphing Linear Equations. There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation. If the equation is given in the form y = mx+b y = m x + b, then m m gives the rise over run value and ...

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1325. Brielle's piggy bank has all nickels and dimes in it. The total value of the money in her piggy bank is $7.80. If Brielle has 66 nickels, write and solve a linear equation to find the number of dimes she has. 45. Direct, Inverse or Neither: y/3 = x. Direct. Direct, Inverse or Neither: xy = 40. Inverse.

What is the equation of the graph in slope intercept form? -1. What is the slope of the equation: y = -x - 14. y = 5/3x -2. What is the equation of the graph in slope intercept form? 5/3. What is the slope of the graph? Study with Quizlet and memorize flashcards containing terms like -2, 7, -3 and more.

Test your understanding of Linear equations, functions, & graphs with these NaN questions. Start test. This topic covers: - Intercepts of linear equations/functions - Slope of linear equations/functions - Slope-intercept, point-slope, & standard forms - Graphing linear equations/functions - Writing linear equations/functions - Interpreting ...

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Example 3.2.10 3.2. 10. Graph: a. x = 2 x = 2 b. y = −1 y = − 1. Solution. a. The equation has only one variable, x, x, and x x is always equal to 2. 2. We create a table where x x is always 2 2 and then put in any values for y. y. The graph is a vertical line passing through the x x -axis at 2. 2.

Linear Equation. An equation of the form Ax + By = C, A x + B y = C, where A and B are not both zero, is called a linear equation in two variables. Here is an example of a linear equation in two variables, x and y. The equation y = −3x + 5 y = −3 x + 5 is also a linear equation. But it does not appear to be in the form Ax + By = C.

This page titled 2.4: Graphing Linear Equations- Answers to the Homework Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform.

This unit is all about understanding linear functions and using them to model real world scenarios. ... Lesson 4 More Work Graphing Linear Functions. LESSON/HOMEWORK. LECCIÓN/TAREA. LESSON VIDEO. ANSWER KEY. EDITABLE LESSON. ... U04.AO.03 - Lesson #9.5 - Absolute Value Equations RESOURCE. ANSWER KEY. EDITABLE RESOURCE. EDITABLE KEY. Add-on ...

Graph a Linear Equation by Plotting Points. There are several methods that can be used to graph a linear equation. The method we used at the start of this section to graph is called plotting points, or the Point-Plotting Method.. Let's graph the equation y = 2 x + 1 y = 2 x + 1 by plotting points.. We start by finding three points that are solutions to the equation.

y = 2x − 4. ... y = mx + b. Graphing linear relationships: equalities inequalities absolute value functions Learn with flashcards, games, and more — for free.

Spencer - Algebra - Unit 4: Inequalities and Linear Equations. Practice Graphing Linear Equations. For each of the following: - Fill in the input/output table for each equation using the domain values. - Graph the solutions, then connect them to create a line. This image cannot currently be displayed. 1. x = y . 2. y = 2x - 3 . 1. 3. 2x + y ...

Common Core Standard: A-REI.C.6, A-REI.D.10, and A-REI.D.12. Student Outcome: Students will solve systems of linear equations graphically, understanding there are three different types of solutions. Students will also use technology (graphing calculator) to solve a system by graphing.

Unit 4: Linear Equations Homework 3: Graphing linear equations. Graphing Linear Equations Lesson & Homework. Recently submitted questions See more. During her work lunch hour, Sharlene went to the mall. Seeing a gorgeous emerald and diamond ring on sale, she pulled out her credit card and purchased it. After work, she returned the ring to the sto

"Name: Unit 4: LInear Equatlons Homework 3: Graphing Linear Equations by Slope-Intercept Form Bell: Date: This is a 2-page document! Directions: Graph the following linear equations: Convert to slope-intercept form when necessary: 3'-3 7+5 V=t+ 7.Y=7-Sx 8. y =-4+5 9.

Graph the linear equation y = 2x + 3 y = 2 x + 3. [hidden-answer a="834421″]Evaluate y = 2x + 3 y = 2 x + 3 for different values of x, and create a table of corresponding x and y values. Convert the table to ordered pairs. Plot the ordered pairs. Draw a line through the points to indicate all of the points on the line.

Description. This Linear Equations Unit Bundle contains guided notes, homework assignments, three quizzes, study guide and a unit test that cover the following topics: • Slope from a Graph. • Slope from Ordered Pairs (The Slope Formula) • Linear Equations: Slope Intercept Form vs. Standard Form.

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Identify the slope and y -intercept of both lines. y = 3 2x + 1 y = 3 2x − 3 y = mx + b y = mx + b m = 3 2 m = 3 2 y-intercept is (0, 1) y-intercept is (0, − 3) The lines have the same slope and different y -intercepts and so they are parallel. You may want to graph the lines to confirm whether they are parallel.

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one way to analyze the relationships between two quantities is to graph the pairs of data on a coordinate axis, graph of a linear equation. in x and y is the set of all points (x,y) that are solutions of the equation, line. graph of a linear equation, x-intercept. x-coordinate of a point where a graph crosses the x-axis,

Question list x/s Question 1 Question 4 Question 5 Question 6 Question 7 Graph the following linear function. Give the domain and range. Identify whether it is a constant function. \[ f(x)=x-11 \] Use the graphing tool to graph the linear equation. Click to enlarge graph Help me solve this View an example Get more help Clear all Check answer