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• Number Theory
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• Everyday Math
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Take a guided, problem-solving based approach to learning Algebra. These compilations provide unique perspectives and applications you won't find anywhere else.

Algebra through Puzzles

What's inside.

• Introduction
• Simplifying Shortcuts
• Arithmetic Logic and Magic
• Balancing Scales
• Rates and Ratios
• Equations and Unknowns
• Manipulating Exponents
• Algebra in Motion
• Common Misconceptions
• Function Fundamentals
• Transformations
• Powers and Radicals
• Polynomials
• Factoring Polynomials

Rational Functions

Piecewise functions, community wiki.

Browse through thousands of Algebra wikis written by our community of experts.

Expressions and Variables

• What Makes A Good Problem?
• Simplifying Expressions
• Distributive Property
• Zero Product Property
• Contest Math I
• Solving Equations
• Setting Up Equations
• Simple Equations
• Verifying Solutions
• Multi-step Equations
• Isolating a Variable
• Balance Puzzles
• System of Linear Equations (Simultaneous Equations)
• How are exponent towers evaluated?
• Does a square root have two values?
• Do Square Roots Always Multiply?
• Is 0.999... = 1?
• What is 0 to the power of 0?
• If AB=AC, does B=C?
• Is (a/b)/c = a/(b/c)?
• Does \(\sqrt{x^2+y^2}=x+y?\)
• Does cross multiply always work for inequalities?
• How does addition in the denominator work?
• List of Common Misconceptions
• Linear Equations
• Forms of Linear Equations
• Equations of Parallel and Perpendicular Lines

Systems of Linear Equations

• Solving Linear Systems Using Matrices
• Absolute Value
• Absolute Value Equations
• Absolute Value Inequalities
• Linear Inequalities
• Absolute Value Linear Inequalities
• System of Inequalities
• Linear Programming

Non-Linear Inequalities

• Polynomial Inequalities
• Classical Inequalities
• Absolute Value Inequalities - 1 Quadratic Term
• Exponential Inequalities
• Logarithmic Inequalities

Quadratic Equations (Parabolas)

• Factoring Quadratics
• Quadratic Equations
• Completing The Square
• Applications of Completing the Square
• Quadratic Discriminant

Square Roots (Radicals)

• Simplifying Radicals
• Radical Equations
• Square Roots
• Rationalizing Denominators

Arithmetic and Geometric Progressions

• Arithmetic Progressions
• Arithmetic Mean
• Harmonic Progression
• Harmonic Mean
• Geometric Progressions
• Geometric Mean
• Arithmetic and Geometric Progressions Problem Solving
• Arithmetic-Geometric Progression
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• Function Composition
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• Function Terminology
• Graphs of Functions
• Transforming Graphs of Functions
• Functional Equations
• Multiplying Polynomials
• Polynomial Division
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• Solving Identity Equations
• Difference Of Squares
• Applying the Perfect Square Identity
• Applying the Perfect Cube Identity
• Factoring by Substitution
• Rational Expressions
• Simplifying Rational Expressions
• Factoring Compound Quadratics: \(\, ax^4 + bx^2 + c\)
• Factoring Cubic Polynomials
• Descartes' Rule of Signs
• Fundamental Theorem of Algebra
• Method of Undetermined Coefficients
• Remainder Factor Theorem
• Transforming Roots of Polynomials
• Intercepts of Rational Functions
• Rational Equations
• Graphing Rational Equations
• Partial Fractions - Linear Factors
• Partial Fractions - Cover Up Rule
• Partial Fractions - Irreducible Quadratics
• Partial Fractions - Repeated Factors
• Telescoping Series - Sum
• Floor Function
• Ceiling Function
• Trailing Number of Zeros
• Fractional Part Function
• Hermite's Identity
• Rules of Exponents
• Simplifying Exponents

Exponential Functions

• Rules of Exponents - Algebraic
• Solving Exponential Equations
• Graphs of Exponential Functions
• Exponential Functions - Problem Solving
• Interest Rate
• Leonhard Euler
• Solving Logarithmic Equations
• Polar Coordinates
• Converting Polar Coordinates to Cartesian
• Parametric Equations
• Complex Numbers
• Complex Conjugates
• Complex Numbers - Absolute Values
• Complex Plane
• Complex Numbers in Geometry
• Gaussian Integers
• Discrete Fourier Transform
• Euler's Formula
• De Moivre's Theorem
• Roots of Unity

Advanced Linear Algebra (Matrices)

• Linear Algebra
• Determinants
• Eigenvalues and Eigenvectors
• Kernel (Nullspace)
• Vector Space
• Cayley-Hamilton Theorem
• Row And Column Spaces
• Spectral Theorem
• Fundamental Subspaces
• Change of Basis
• Rank-Nullity Theorem
• Linear Transformations
• Linear Independence
• Jordan Canonical Form
• Affine transformations
• Vieta Root Jumping

Advanced Polynomials

• Algebraic Manipulation
• Algebraic Manipulation Identities
• Sum of n, n², or n³
• Telescoping Series - Product
• Nested Functions
• Cardano's Method
• Cubic Discriminant
• Gauss: The Prince of Mathematics
• Srinivasa Ramanujan
• Binomial Coefficient
• Pascal's Triangle
• Binomial Theorem
• Negative Binomial Theorem
• Fractional Binomial Theorem
• Rational Root Theorem
• Vieta's Formula
• Newton's Identities
• Completing the Square - Multiple Variables
• Algebraic Identities
• Factorization of Polynomials
• Sophie Germain Identity
• Algebraic Number Theory
• Fermat's Last Theorem
• Eisenstein's Irreducibility Criterion
• Group Theory
• Lagrange's Theorem
• Schwartz-Zippel Lemma
• Polynomial Interpolation by Remainder Factor Theorem
• Lagrange Interpolation
• Method of Differences
• Primitive Roots of Unity
• Symmetric Polynomials
• The \(uvw\) Method
• Chebyshev Polynomials - Definition and Properties

Advanced Inequalities

• Cauchy-Schwarz Inequality
• Titu's Lemma
• Rearrangement Inequality
• Chebyshev's Inequality
• Jensen's Inequality
• Hölder's Inequality
• Young's Inequality
• Muirhead Inequality
• Reverse Rearrangement Inequality
• Arithmetic Mean - Geometric Mean
• Applying the Arithmetic Mean Geometric Mean Inequality
• Power Mean Inequality (QAGH)
• Quadratic Mean
• Triangle Inequality
• AIME Math Contest Preparation
• JEE Exam Preparation
• RMO Math Contest Preparation
• INMO Math Contest Preparation
• KVPY Exam Preparation
• BITSAT Exam Preparation

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Simple Algebra Problems – Easy Exercises with Solutions for Beginners

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Understanding Algebraic Expressions

Breaking down algebra problems, solving algebraic equations, tackling algebra word problems, types of algebraic equations, algebra for different grades.

For instance, solving the equation (3x = 7) for (x) helps us understand how to isolate the variable to find its value.

I always find it fascinating how algebra serves as the foundation for more advanced topics in mathematics and science. Starting with basic problems such as ( $(x-1)^2 = [4\sqrt{(x-4)}]^2$ ) allows us to grasp key concepts and build the skills necessary for tackling more complex challenges.

So whether you’re refreshing your algebra skills or just beginning to explore this mathematical language, let’s dive into some examples and solutions to demystify the subject. Trust me, with a bit of practice, you’ll see algebra not just as a series of problems, but as a powerful tool that helps us solve everyday puzzles.

Simple Algebra Problems and Strategies

When I approach simple algebra problems, one of the first things I do is identify the variable.

The variable is like a placeholder for a number that I’m trying to find—a mystery I’m keen to solve. Typically represented by letters like ( x ) or ( y ), variables allow me to translate real-world situations into algebraic expressions and equations.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like ( x ) or ( y )), and operators (like add, subtract, multiply, and divide). For example, ( 4x + 7 ) is an algebraic expression where ( x ) is the variable and the numbers ( 4 ) and ( 7 ) are terms. It’s important to manipulate these properly to maintain the equation’s balance.

Solving algebra problems often starts with simplifying expressions. Here’s a simple method to follow:

• Combine like terms : Terms that have the same variable can be combined. For instance, ( 3x + 4x = 7x ).
• Isolate the variable : Move the variable to one side of the equation. If the equation is ( 2x + 5 = 13 ), my job is to get ( x ) by itself by subtracting ( 5 ) from both sides, giving me ( 2x = 8 ).

With algebraic equations, the goal is to solve for the variable by performing the same operation on both sides. Here’s a table with an example:

Algebra word problems require translating sentences into equations. If a word problem says “I have six less than twice the number of apples than Bob,” and Bob has ( b ) apples, then I’d write the expression as ( 2b – 6 ).

Understanding these strategies helps me tackle basic algebra problems efficiently. Remember, practice makes perfect, and each problem is an opportunity to improve.

In algebra, we encounter a variety of equation types and each serves a unique role in problem-solving. Here, I’ll brief you about some typical forms.

Linear Equations : These are the simplest form, where the highest power of the variable is one. They take the general form ( ax + b = 0 ), where ( a ) and ( b ) are constants, and ( x ) is the variable. For example, ( 2x + 3 = 0 ) is a linear equation.

Polynomial Equations : Unlike for linear equations, polynomial equations can have variables raised to higher powers. The general form of a polynomial equation is ( $a_nx^n + a_{n-1}x^{n-1} + … + a_2x^2 + a_1x + a_0 = 0$ ). In this equation, ( n ) is the highest power, and ( $a_n$ ), ( $a_{n-1} $), …, ( $a_0$ ) represent the coefficients which can be any real number.

• Binomial Equations : They are a specific type of polynomial where there are exactly two terms. Like ($ x^2 – 4 $), which is also the difference of squares, a common format encountered in factoring.

To understand how equations can be solved by factoring, consider the quadratic equation ( $x^2$ – 5x + 6 = 0 ). I can factor this into ( (x-2)(x-3) = 0 ), which allows me to find the roots of the equation.

Here’s how some equations look when classified by degree:

Remember, identification and proper handling of these equations are essential in algebra as they form the basis for complex problem-solving.

In my experience with algebra, I’ve found that the journey begins as early as the 6th grade, where students get their first taste of this fascinating subject with the introduction of variables representing an unknown quantity.

I’ve created worksheets and activities aimed specifically at making this early transition engaging and educational.

6th Grade :

Moving forward, the complexity of algebraic problems increases:

7th and 8th Grades :

• Mastery of negative numbers: students practice operations like ( -3 – 4 ) or ( -5 $\times$ 2 ).
• Exploring the rules of basic arithmetic operations with negative numbers.
• Worksheets often contain numeric and literal expressions that help solidify their concepts.

Advanced topics like linear algebra are typically reserved for higher education. However, the solid foundation set in these early grades is crucial. I’ve developed materials to encourage students to understand and enjoy algebra’s logic and structure.

Remember, algebra is a tool that helps us quantify and solve problems, both numerical and abstract. My goal is to make learning these concepts, from numbers to numeric operations, as accessible as possible, while always maintaining a friendly approach to education.

I’ve walked through various simple algebra problems to help establish a foundational understanding of algebraic concepts. Through practice, you’ll find that these problems become more intuitive, allowing you to tackle more complex equations with confidence.

Remember, the key steps in solving any algebra problem include:

• Identifying variables and what they represent.
• Setting up the equation that reflects the problem statement.
• Applying algebraic rules such as the distributive property ($a(b + c) = ab + ac$), combining like terms, and inverse operations.
• Checking your solutions by substituting them back into the original equations to ensure they work.

As you continue to engage with algebra, consistently revisiting these steps will deepen your understanding and increase your proficiency. Don’t get discouraged by mistakes; they’re an important part of the learning process.

I hope that the straightforward problems I’ve presented have made algebra feel more manageable and a little less daunting. Happy solving!

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• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
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Algebra Problems

Algebra problems are not only based on algebraic expressions but also on various types of equations in Maths where a quantity or variable is unknown to us. Many of us are familiar with the word problem, but are we aware of the fact and problems related to variables and constants? When we say 5 it means a number but what if we say x=5 or 5y or something like that?

This is where algebra came into existence algebra is that branch of mathematics which not only deals with numbers but also variable and alphabets. The versatility of Algebra is very deep and very conceptual, all the non-numeric character represents variable and numeric as constants. Let us solve some problems based algebra with solutions which will cover the syllabus for class 6, 7, 8. Below are some of the examples of algebraic expressions .

For example.

Algebra Word Problems deal with real-time situations and solutions which can be solved using algebra.

Basic Algebra Identities

• (a + b) 2  = a 2 + b 2 + 2ab
• (a – b) 2 = a 2  + b 2 – 2ab
• a 2 – b 2 = (a + b)(a – b)
• a 2 + b 2 = (a + b) 2 – 2ab = (a – b) 2 + 2ab
• a 3 + b 3 = (a + b)(a 2 – ab + b 2 )
• a 3 – b 3 = (a – b)(a 2 + ab + b 2 )
• (a + b) 3 = a 3 + 3ab(a + b) + b 3
• (a – b) 3 = a 3 – 3ab(a – b) – b 3

Algebra problems With Solutions

Example 1: Solve, (x-1) 2 = [4√(x-4)] 2 Solution: x 2 -2x+1 = 16(x-4)

x 2 -2x+1 = 16x-64

x 2 -18x+65 = 0

(x-13) (x-5) = 0

Hence, x = 13 and x = 5.

Algebra Problems for Class 6

In class 6, students will be introduced with an algebra concept. Here, you will learn how the unknown values are represented in terms of variables.  The given expression can be solved only if we know the value of unknown variable. Let us see some examples.

Example: Solve, 4x + 5 when, x = 3.

Solution: Given, 4x + 5

Now putting the value of x=3, we get;

4 (3) + 5 = 12 + 5 = 17.

Example: Give expressions for the following cases:

(i) 12 added to 2x

(ii) 6 multiplied by y

(iii) 25 subtracted from z

(iv) 17 times of m 

(i) 12 + 2x

Algebra Problems for Class 7

In class 7, students will deal with algebraic expressions like x+y, xy, 32x 2 -12y 2 , etc. There are different kinds of the terminology used in case algebraic equations such as;

• Coefficient

Let us understand these terms with an example. Suppose 4x + 5y is an algebraic expression, then 4x and 5y are the terms. Since here the variables used are x and y, therefore, x and y are the factors of 4x + 5y. And the numerical factor attached to the variables are the coefficient such as 4 and 5 are the coefficient of x and y in the given expression.

Any expression with one or more terms is called a polynomial. Specifically, a one-term expression is called a monomial; a two-term expression is called a binomial, and a three-term expression is called a trinomial.

Terms which have the same algebraic factors are like terms . Terms which have different algebraic factors are unlike terms . Thus, terms 4xy and – 3xy are like terms; but terms 4xy and – 3x are not like terms.

Example: Add 3x + 5x

Solution: Since 3x and 5x have the same algebraic factors, hence, they are like terms and can be added by their coefficient.

3x + 5x = 8x

Example: Collect like terms and simplify the expression: 12x 2 – 9x + 5x – 4x 2 – 7x + 10.

Solution: 12x 2 – 9x + 5x – 4x 2 – 7x + 10

= (12 – 4)x 2 – 9x + 5x – 7x + 10

=  8x 2 – 11x + 10

Algebra Problems for Class 8

Here, students will deal with algebraic identities. See the examples.

Example: Solve (2x+y) 2

Solution: Using the identity: (a+b) 2  = a 2 + b 2 + 2 ab, we get;

(2x+y) = (2x) 2 + y 2 + 2.2x.y = 4x 2 + y 2 + 4xy

Example: Solve (99) 2 using algebraic identity.

Solution: We can write, 99 = 100 -1

Therefore, (100 – 1 ) 2

= 100 2 + 1 2 – 2 x 100 x 1  [By identity: (a -b) 2 = a 2 + b 2 – 2ab

= 10000 + 1 – 200

Algebra Word Problems

Question 1: There are 47 boys in the class. This is three more than four times the number of girls. How many girls are there in the class?

Solution: Let the number of girls be x

As per the given statement,

4 x + 3 = 47

4x = 47 – 3

Question 2: The sum of two consecutive numbers is 41. What are the numbers?

Solution: Let one of the numbers be x.

Then the other number will x+1

Now, as per the given questions,

x + x + 1 = 41

2x + 1 = 41

So, the first number is 20 and second number is 20+1 = 21

Linear Algebra Problems

There are various methods For Solving the Linear Equations

• Cross multiplication method
• Replacement method or Substitution method
• Hit and trial method

There are Variety of different Algebra problem present and are solved depending upon their functionality and state. For example, a linear equation problem can’t be solved using a quadratic equation formula and vice verse for, e.g., x+x/2=7 then solve for x is an equation in one variable for x which can be satisfied by only one value of x. Whereas x 2 +5x+6 is a quadratic equation which is satisfied for two values of x the domain of algebra is huge and vast so for more information. Visit BYJU’S. where different techniques are explained different algebra problem.

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Praxis Core Math

Course: praxis core math   >   unit 1.

• Algebraic properties | Lesson
• Algebraic properties | Worked example
• Solution procedures | Lesson
• Solution procedures | Worked example
• Equivalent expressions | Lesson
• Equivalent expressions | Worked example
• Creating expressions and equations | Lesson
• Creating expressions and equations | Worked example

Algebraic word problems | Lesson

• Algebraic word problems | Worked example
• Linear equations | Lesson
• Linear equations | Worked example
• Quadratic equations | Lesson
• Quadratic equations | Worked example

What are algebraic word problems?

What skills are needed.

• Translating sentences to equations
• Solving linear equations with one variable
• Evaluating algebraic expressions
• Solving problems using Venn diagrams

How do we solve algebraic word problems?

• Define a variable.
• Write an equation using the variable.
• Solve the equation.
• If the variable is not the answer to the word problem, use the variable to calculate the answer.

What's a Venn diagram?

• 7 + 10 − 13 = 4 ‍   brought both food and drinks.
• 7 − 4 = 3 ‍   brought only food.
• 10 − 4 = 6 ‍   brought only drinks.
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
• (Choice A)   $ 4 ‍   A $ 4 ‍  
• (Choice B)   $ 5 ‍   B $ 5 ‍  
• (Choice C)   $ 9 ‍   C $ 9 ‍  
• (Choice D)   $ 14 ‍   D $ 14 ‍  
• (Choice E)   $ 20 ‍   E $ 20 ‍  
• (Choice A)   10 ‍   A 10 ‍  
• (Choice B)   12 ‍   B 12 ‍  
• (Choice C)   24 ‍   C 24 ‍  
• (Choice D)   30 ‍   D 30 ‍  
• (Choice E)   32 ‍   E 32 ‍  
• (Choice A)   4 ‍   A 4 ‍  
• (Choice B)   10 ‍   B 10 ‍  
• (Choice C)   14 ‍   C 14 ‍  
• (Choice D)   18 ‍   D 18 ‍  
• (Choice E)   22 ‍   E 22 ‍  

Things to remember

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