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Defining variables

Variables and expressions.

Students define variables and write expressions to answer questions with unknown amounts. These worksheets demonstrate the need to clearly define what a variable represents.

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Algebra Worksheets

Welcome to the Algebra worksheets page at Math-Drills.com, where unknowns are common and variables are the norm. On this page, you will find Algebra worksheets mostly for middle school students on algebra topics such as algebraic expressions, equations and graphing functions.

This page starts off with some missing numbers worksheets for younger students. We then get right into algebra by helping students recognize and understand the basic language related to algebra. The rest of the page covers some of the main topics you'll encounter in algebra units. Remember that by teaching students algebra, you are helping to create the future financial whizzes, engineers, and scientists that will solve all of our world's problems.

Algebra is much more interesting when things are more real. Solving linear equations is much more fun with a two pan balance, some mystery bags and a bunch of jelly beans. Algebra tiles are used by many teachers to help students understand a variety of algebra topics. And there is nothing like a set of co-ordinate axes to solve systems of linear equations.

Most Popular Algebra Worksheets this Week

Properties and Laws of Numbers Worksheets

The commutative law.

The commutative law or commutative property states that you can change the order of the numbers in an arithmetic problem and still get the same results. In the context of arithmetic, it only works with addition or multiplication operations , but not mixed addition and multiplication. For example, 3 + 5 = 5 + 3 and 9 × 5 = 5 × 9. A fun activity that you can use in the classroom is to brainstorm non-numerical things from everyday life that are commutative and noncommutative. Putting on socks, for example, is commutative because you can put on the right sock then the left sock or you can put on the left sock then the right sock and you will end up with the same result. Putting on underwear and pants, however, is noncommutative.

The Associative Law

The associative law or associative property allows you to change the grouping of the operations in an arithmetic problem with two or more steps without changing the result. The order of the numbers stays the same in the associative law. As with the commutative law, it applies to addition-only or multiplication-only problems. It is best thought of in the context of order of operations as it requires that parentheses must be dealt with first. An example of the associative law is: (9 + 5) + 6 = 9 + (5 + 6). In this case, it doesn't matter if you add 9 + 5 first or 5 + 6 first, you will end up with the same result. Students might think of some examples from their experience such as putting items on a tray at lunch. They could put the milk and vegetables on their tray first then the sandwich or they could start with the vegetables and sandwich then put on the milk. If their tray looks the same both times, they will have modeled the associative law. Reading a book could be argued as either associative or nonassociative as one could potentially read the final chapters first and still understand the book as well as someone who read the book the normal way.

Inverse relationships with one blank

Inverse relationships worksheets cover a pre-algebra skill meant to help students understand the relationship between multiplication and division and the relationship between addition and subtraction.

Inverse relationships with two blanks

Missing Numbers or Unknowns in Equations Worksheets

Missing numbers in equations worksheets in three types: blanks for unknowns, symbols for unknowns and variables for unknowns.

Missing Numbers Worksheets with Blanks as Unknowns

In these worksheets, the unknown is limited to the question side of the equation which could be on the left or the right of equal sign.

Missing Numbers Worksheets with Symbols as Unknowns

Equalities with addition on both sides of the equation and symbols as unknowns

Missing numbers worksheets with variables as unknowns

Solving Simple Linear Equations

Algebraic Expressions Worksheets

Using the distributive property.

The distributive property is an important skill to have in algebra. In simple terms, it means that you can split one of the factors in multiplication into addends, multiply each addend separately, add the results, and you will end up with the same answer. It is also useful in mental math, and example of which should help illustrate the definition. Consider the question, 35 × 12. Splitting the 12 into 10 + 2 gives us an opportunity to complete the question mentally using the distributive property. First multiply 35 × 10 to get 350. Second, multiply 35 × 2 to get 70. Lastly, add 350 + 70 to get 420. In algebra, the distributive property becomes useful in cases where one cannot easily add the other factor before multiplying. For example, in the expression, 3(x + 5), x + 5 cannot be added without knowing the value of x. Instead, the distributive property can be used to multiply 3 × x and 3 × 5 to get 3x + 15.

Evaluating algebraic expressions

Exponent Rules and Properties

Practice with basic exponent rules.

As the title says, these worksheets include only basic exponent rules questions. Each question only has two exponents to deal with; complicated mixed up terms and things that a more advanced student might work out are left alone. For example, 4 2 is (2 2 ) 2 = 2 4 , but these worksheets just leave it as 4 2 , so students can focus on learning how to multiply and divide exponents more or less in isolation.

Linear Expressions & Equations

Linear equations worksheets including simplifying, graphing, evaluating and solving systems of linear equations.

Translating algebraic phrases in words to algebraic expressions

Knowing the language of algebra can help to extract meaning from word problems and to situations outside of school. In these worksheets, students are challenged to convert phrases into algebraic expressions.

Simplifying linear expressions (combining like terms)

Combining like terms is something that happens a lot in algebra. Students can be introduced to the topic and practice a bit with these worksheets. The bar is raised with the adding and subtracting versions that introduce parentheses into the expressions. For students who have a good grasp of fractions, simplifying simple algebraic fractions worksheets present a bit of a challenge over the other worksheets in this section.

Rewriting linear equations

Determining linear equations from slopes, y-intercepts, and points

Linear Equation Graphs

Graphing linear equations and reading existing graphs give students a visual representation that is very useful in understanding the concepts of slope and y-intercept.

Solving linear equations with jelly beans is a fun activity to try with students first learning algebraic concepts. Ideally, you will want some opaque bags with no mass, but since that isn't quite possible (the no mass part), there is a bit of a condition here that will actually help students understand equations better. Any bags that you use have to be balanced on the other side of the equation with empty ones.

Probably the best way to illustrate this is through an example. Let's use 3 x + 2 = 14. You may recognize the x as the unknown which is actually the number of jelly beans we put in each opaque bag. The 3 in the 3 x means that we need three bags. It's best to fill the bags with the required number of jelly beans out of view of the students, so they actually have to solve the equation.

On one side of the two-pan balance, place the three bags with x jelly beans in each one and two loose jelly beans to represent the + 2 part of the equation. On the other side of the balance, place 14 jelly beans and three empty bags which you will note are required to "balance" the equation properly. Now comes the fun part... if students remove the two loose jelly beans from one side of the equation, things become unbalanced, so they need to remove two jelly beans from the other side of the balance to keep things even. Eating the jelly beans is optional. The goal is to isolate the bags on one side of the balance without any loose jelly beans while still balancing the equation.

The last step is to divide the loose jelly beans on one side of the equation into the same number of groups as there are bags. This will probably give you a good indication of how many jelly beans there are in each bag. If not, eat some and try again. Now, we realize this won't work for every linear equation as it is hard to have negative jelly beans, but it is another teaching strategy that you can use for algebra.

Solving linear equations

Despite all appearances, equations of the type a/ x are not linear. Instead, they belong to a different kind of equations. They are good for combining them with linear equations, since they introduce the concept of valid and invalid answers for an equation (what will be later called the domain of a function). In this case, the invalid answers for equations in the form a/ x , are those that make the denominator become 0.

Linear Systems

Solving systems of linear equations.

Solving systems of linear equations by graphing

Quadratic Expressions & Equations

Quadratic expressions and equations worksheets including multiplying factors, factoring, and solving quadratic equations.

Simplifying quadratic expressions (combining like terms)

Adding/Subtracting and Simplifying quadratic expressions

Multiplying factors of quadratic expressions

Factoring quadratic expressions

The factoring quadratic expressions worksheets in this section provide many practice questions for students to hone their factoring strategies. If you would rather worksheets with quadratic equations, please see the next section. These worksheets come in a variety of levels with the easier ones are at the beginning. The 'a' coefficients referred to below are the coefficients of the x 2 term as in the general quadratic expression: ax 2 + bx + c. There are also worksheets in this section for calculating sum and product and for determining the operands for sum and product pairs.

Whether you use trial and error, completing the square or the general quadratic formula, these worksheets include a plethora of practice questions with answers. In the first section, the worksheets include questions where the quadratic expressions equal 0. This makes the process similar to factoring quadratic expressions, with the additional step of finding the values for x when the expression is equal to 0. In the second section, the expressions are generally equal to something other than x, so there is an additional step at the beginning to make the quadratic expression equal zero.

Solving Quadratic equations that Equal Zero (e.g. ax² + bx + c = 0)

Solving Quadratic equations that Equal an Integer (e.g. ax² + bx + c = d)

Other Polynomial and Monomial Expressions & Equations

Factoring non-quadratic expressions worksheets with various levels of complexity.

Simplifying polynomials that involve addition and subtraction

Simplifying polynomials that involve multiplication and division

Simplifying polynomials that involve addition, subtraction, multiplication and division

Factoring expressions that do not include a squared variable

Factoring expressions that always include a squared variable

Factoring expressions that sometimes include squared variables

Multiplying polynomials with two factors

Multiplying polynomials with three factors

Inequalities Including Graphs

Inequalities worksheets including writing the inequality that matches a graph and graphing inequalities on a number line.

Writing the inequality that matches the graph

Graphing inequalities on number lines

Solving linear inequalities

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Variable Worksheets

• Kindergarten

Variable Expressions Worksheets

Variable expressions worksheets have mathematical phrases that connect numbers, variables, and operators for showing the value of something. It represents the value of something. variable expressions worksheets help students to be evaluated. equations can be solved and often they make two expressions equal.

Benefits of Variable Expressions Worksheets

Variable expressions worksheets can control the number of problems, difficulty level, range of numbers used, workspace below the problems, border around the problems, and additional instructions.

Variable expressions worksheets come with visual simulation for students to see the problems in action, and provide a detailed step-by-step solution for students to understand the process better. Students get equipped with the problems of variables with the help of these worksheets.

variable expressions worksheets are well-curated to enhance a child's learnability through interactive exercises and problems based on the order of operations. By practicing multiple problems covered in these worksheets, a child can quickly grasp both easy and complicated concepts with ease.

Download Variable Expressions Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

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Printable Algebra Worksheets

Algebra is a branch of math in which letters and symbols are used to represent numbers and quantities in formulas and equations. The assemblage of printable algebra worksheets encompasses topics like translating phrases, evaluating and simplifying algebraic expressions, solving equations, graphing linear and quadratic equations, comprehending linear and quadratic functions, inequalities, polynomials, trigonometry, calculus and much more.

List of Algebra Worksheets

• Translating Algebraic Expressions
• Solving Equations
• Equation Word Problems
• Rearranging Equations
• Equation of a Line / Linear Equations
• Graphing Linear Equations
• Graphing Linear Functions
• Transformation of a Linear Functions
• Quadratic Functions
• Graphing Quadratic Functions
• Quadratic Transformation
• Sequence and series
• Determinants

Explore the Algebra Worksheets in Detail

Translating Algebraic Expressions Worksheets

The algebra worksheets here focus on translating verbal phrases to algebraic expression featuring exercises with single and multivariable expressions, translating linear equations involving one or two-steps, determining inequalities and much more.

Evaluating Algebraic Expressions Worksheets

Access the printable worksheets here to practice evaluation of algebraic expressions with single and multivariables, find the dimensions of geometric shapes, arrange the algebraic expressions in ascending or descending order to mention a few.

Simplifying Algebraic Expressions Worksheets

Enrich your knowledge in simplifying algebraic expressions with this batch of worksheets, containing skills to simplify linear, polynomial and rational expressions; expressions involving positive and negative exponents, finding the area and perimeter of a rectangle and more.

Algebraic Identities Worksheets

Practice this collection of algebraic identities worksheets that feature vibrant identities charts, and hone your skills in expanding, factoring and evaluating algebraic expression using identities, simplifying the expressions and much more.

Solving Equation Worksheets

Choose from an inexhaustible range of worksheets consisting of one-step, two-step and multi-step equations. Write the equation of a line in varied forms, graph linear equations, quadratic equations and absolute value equations, solve the system of equations, to mention a few.

Equation Word Problems Worksheets

Review the concept of solving equations with these equation word problems worksheets. Solve real-life word problems featuring integers, decimals and fractions involving one-step, two-steps or multi-steps.

Rearranging Equation Worksheets

Incorporate these rearranging equation worksheets featuring skills to make 'x' the subject, rearrange and evaluate frequently used formulae and solve real word problems available in customary and metric units and more.

Equation of a Line / Linear Equation Worksheets

Employ the set of algebra worksheets here to find the linear equation of a line using the point-slope form, slope-intercept form, two-point form, two-intercept form. Also, find the x-intercept and y-intercept, solve word problems involving parallel and perpendicular lines to mention a few.

Graphing Linear Equation Worksheets

Obtain umpteen worksheets featuring completion of function tables, plotting a graph using slope and y-intercept, graphing equations involving horizontal and vertical lines and more. Included here are worksheets to plot ordered pairs as well.

Quadratic Equation Worksheets

Implement this set of quadratic equation worksheets to solve equations by finding the sum and product of the roots, usage of the zero product property, factorization method, and quadratic formula. Additionally, learn to solve by completing the square as well.

Function Worksheets

This unit specifically deals with exercises to find the domain and range from the list of ordered pairs and graphs. Learn to complete function tables; plot points, graph functions and evaluate the composition of functions.

Graphing Linear Function Worksheets

This ensemble consists of meticulously on graphing linear functions by plotting points on grids, computing the function tables, plotting points and graphing linear functions. Included here are slopes presented as fractions as well.

Transformation of a Linear Function Worksheets

This batch of algebra worksheets encompasses tasks for a clear understanding of the transformation of a linear function and graph. Translate a function or a graph to horizontal/vertical shift, find the reflection, find the stretch and compression and more.

Quadratic Function Worksheets

Implement this set of tailor-made worksheets with ample exercises to evaluate or write quadratic functions; interpreting the quadratic function in varied forms, completing function tables, identifying the vertex and intercepts based on formulae and a lot more.

Graphing Quadratic Function Worksheets

Identify zeroes, write and graph quadratic functions, complete the function table are some of the exercises included in this compilation of practice printables.

Quadratic Transformation Worksheets

Utilize this unit of simple yet engaging quadratic transformation worksheets to transform graphs, find the transformation function g(x) from its parent f(x) and to identify the various types of shifts to mention a few.

Polynomial Worksheets

Access these algebra worksheets to recognize polynomials, identify the degree of polynomials, add, subtract, multiply, divide and factorize monomial, binomial and polynomial expressions. Find the LCM and GCF of polynomials as well.

Absolute Value Worksheets

Navigate through this assortment of printable absolute value worksheets with positive and negative integers. Perform arithmetic operations like addition, subtraction, multiplication and division on the absolute value of real numbers.

Inequalities Worksheets

Avail this section of inequality worksheets to graph and write inequalities, solve one-step, two-step or multi-step inequalities, solve and graph compound inequalities, graph solutions and much more.

Sequence and Series Worksheets

Supplement your core instructions with this group of sequence and series worksheets focusing on arithmetic series and sequences, geometric sequence, special series, recursive sequence, partial sum to mention a few.

Matrices Worksheets

Achieve optimal learning with this set of matrix worksheets. Determine the order of matrices and their inverse; add, subtract and multiply matrices; find their solutions and determinants as well in this assemblage of printable worksheets here.

Determinants Worksheets

Gain in-depth knowledge in finding the determinants; identifying the number of solutions and solving equations using Cramer's rule with this batch of worksheets containing innumerable exercises.

Complex Numbers Worksheets

Explore this compilation of complex number worksheets designed and recommended for high-school students. Hone your skills in finding the absolute value and argument, simplifying, evaluating powers of i, finding the conjugate, identifying the real and imaginary part and more!

Vectors Worksheets

This stock of printable vector worksheets is an ideal resource for children to practice representing vectors in component form and polar form; adding, subtracting, and multiplying vectors; finding unit vectors, resultant vectors, and the magnitude and direction angle of a vector; and many more.

Trigonometry Worksheets

Browse through umpteen trigonometric worksheets dealing with the primary, reciprocal trigonometric ratios, quadrants and angles, reference and coterminal angles, Pythagorean theorem worksheets and a lot more.

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Variables in Expressions and Equations Worksheets

We begin to real spend a good bit of time chasing after those pesky unknown values in math expressions and equations called variables. These little placeholders are often signified through use of letters as symbols. The two most commonly used letters are x and y. We need to remember that any symbol can be used to define a variable as long as it is consistently used throughout the problem. I had a College Physics professor that was known for using the most unique symbols to define variables. I remember that we were trying to determine the speed at which a football travelled on the school field. He used a car with screeching tires to define speed, a meter stick to define distance, and a clock to define time. These worksheets and lessons will help students understand the purpose and uses of variable in equations and expressions.

Aligned Standard: Grade 6 Expressions and Equations - 6.EE.B.6

• Variables In Word Problems Step-by-step Lesson - We give you a whole paragraph to sift through. We give you the variables and ask you to complete the thought.
• Guided Lesson - Write expressions from word problems and solve a premade expression.
• Guided Lesson Explanation - It will really help you, if you outline the thoughts behind this one before you proceed with it.
• Practice Worksheet - The variables are presented in the word problems and we ask you to solve them midstream.
• Matching Worksheet - Match the word problems to the expressions.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

These problems are all about find differences in a situation.

• Homework 1 - Max went to the market. There he saw 7 puppies. X puppies lost their way. 5 puppies were left at the market. How many puppies lost their way?
• Homework 2 - There were two fish ponds. Each pond has 50 fish. But f fish die due to environmental pollution. Now only 24 fish are left. How many fish past away?
• Homework 3 - A company ships 10 boxes of flip flops across the country. The total number of pairs of flip flops was 100. How many pairs of flip flops did the company put in each box?

Practice Worksheets

The use of variables here is directly implied to help students grasp the concept.

• Practice 1 - There were two packs of match boxes. Each box has 100 match sticks. But m match sticks were damaged. Now only 67 sticks are left. How many total sticks were damaged?
• Practice 2 - Winnie has 10 kitties. Nina gives her d more kitties. Write the expression that shows the total number of kitties Winnie has.
• Practice 3 - Ed has 8 burgers. He ate x numbers of burgers. Write the expression that shows the number of burgers left.

Math Skill Quizzes

Each quiz tends to focus on one area of the skill. Make sure to look at them all.

• Quiz 1 - Write the expression that shows p is added to 40?
• Quiz 2 - Write the expression that shows 650 is subtracted from m.
• Quiz 3 - Joy has 6 lollipops. Matt gives her d more lollipops. Write the expression that shows the number of lollipops Joy has.

Why Would You Use Variables to Represent Numbers?

You will find variables (symbols designated most often by letters) in many mathematics topics, so understanding why they are used in the first place is essential. Variables are those letter symbols that are used to represent unknown values in a problem. It often seems like the more difficult a problem becomes; the more unknown numbers that are present.

Variables allow us to state that we have an unknown and to specify where it lies within the expression, statement, equation, or inequality. While that does not seem like it carries a lot of weight, it is the perfect place to define a set unknown value because it allows us to track it and determine the value, if we so choose to do so. With the help of variables, you can easily convert a word problem into an equation, which is the first step for solving it. Let us consider an example:

12x + 3 = 8x + 7

Here, you need to find the value of x, which makes the above equation true.

So, the first step consists of determining the value of the variable (with its numbers) on one side of the equal sign. While the constants (numbers) should be placed on the other side of the equal sign.

By changing the sign, the signs before moving variable or number will be changed oppositely. That means, here the plus sign will change to the minus sign.

12x - 8x = 7 - 3

Now, we will simplify the above equation. 4x = 4. This is where it gets tricky; we know that we want the value of x and not 4x. So, we want to keep x alone on one side and place the constants on the other. Now, the number with the variable is basically multiplied to it. So, when you take 4 on the other side then, it will change from a multiplying number to a dividing number. Let us look at the example.

x = 4/4. Now, 4 divided by 4 is equal to 1. So, the value of x is 1. x = 1.

In algebra, variables are the primary quantity that we are often spending most of our time determining. Understanding how to properly rearrange equations, inequalities, and expressions is often more than half the battle. If we think of variables as placeholders, it often makes it very simple to work through. Just as the name indicates, the end value of variables can vary based on the situation that they find themselves in.

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Key to Algebra Workbooks

Key to Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language, and examples are easy to follow. Word problems relate algebra to familiar situations, helping students to understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system.

Free Printable Math Worksheets for Algebra 1

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• Writing variable expressions
• Order of operations
• Evaluating expressions
• Number sets
• Adding rational numbers
• Adding and subtracting rational numbers
• Multiplying and dividing rational numbers
• The distributive property
• Combining like terms
• Percent of change
• One-step equations
• Two-step equations
• Multi-step equations
• Absolute value equations
• Solving proportions
• Percent problems
• Distance-rate-time word problems
• Mixture word problems
• Work word problems
• Literal Equations
• Graphing one-variable inequalities
• One-step inequalities
• Two-step inequalities
• Multi-step inequalities
• Compound inequalities
• Absolute value inequalities
• Discrete relations
• Continuous relations
• Evaluating and graphing functions
• Finding slope from a graph
• Finding slope from two points
• Finding slope from an equation
• Graphing lines using slope-intercept form
• Graphing lines using standard form
• Writing linear equations
• Graphing linear inequalities
• Graphing absolute value equations
• Direct variation
• Solving systems of equations by graphing
• Solving systems of equations by elimination
• Solving systems of equations by substitution
• Systems of equations word problems
• Graphing systems of inequalities
• Discrete exponential growth and decay word problems
• Exponential functions and graphs
• Writing numbers in scientific notation
• Operations with scientific notation
• Addition and subtraction with scientific notation
• Naming polynomials
• Adding and subtracting polynomials
• Multiplying polynomials
• Multiplying special case polynomials
• Factoring special case polynomials
• Factoring by grouping
• Dividing polynomials
• Graphing quadratic inequalities
• Completing the square
• By taking square roots
• By factoring
• With the quadratic formula
• By completing the square
• Simplifying radicals
• Adding and subtracting radical expressions
• Multiplying radicals
• Dividing radicals
• Using the distance formula
• Using the midpoint formula
• Simplifying rational expressions
• Finding excluded values / restricted values
• Multiplying rational expressions
• Dividing rational expressions
• Adding and subtracting rational expressions
• Finding trig. ratios
• Finding angles of triangles
• Finding side lengths of triangles
• Visualizing data
• Center and spread of data
• Scatter plots
• Using statistical models