Solving One-Step Equations

Introduction

This Grades 7–8 course focuses on a core algebra skill. A one-step equation is an equation that you can solve by using one inverse operation to isolate the unknown. In this course, you will work mainly with one-variable linear equations such as $x + 7 = 15$ , $4 x = 28$ , and $\frac{x}{5} = - 3$ . These equations are short, but they teach ideas that are essential for later work with Algebra, inequalities, formulas, functions, and word problems.

The most important idea is that an equation describes two expressions with the same value. The equals sign does not mean "the answer comes next." It means "the expression on the left has the same value as the expression on the right."

You will learn to choose the correct inverse operation, apply it consistently, explain why the equation stays balanced, and check your result by substitution. You will also translate short real-world situations into equations and decide whether a proposed solution is reasonable.

Learning Goals

By the end of this aiMOOC, you should be able to explain what a variable and a solution mean, solve one-step equations involving addition, subtraction, multiplication, and division, work confidently with negative numbers, fractions, and decimals, check solutions by substitution, model simple situations with equations, and explain your reasoning using the balance principle.

Understanding Equations

Variables, Constants, and Solutions

A variable is a symbol, usually a letter, that represents an unknown or changeable number. In $x + 6 = 14$ , the variable is $x$ . The number 6 is a constant. A solution is a value of the variable that makes the equation true.

For example, $x = 8$ is the solution of $x + 6 = 14$ because substituting 8 gives $8 + 6 = 14$ , which is true. The goal of solving is not simply to move symbols around. The goal is to find every value that makes the original equation true.

The Equality Principle

Think of an equation as a balanced scale. If both sides have the same value, the scale is balanced. To preserve the equality while solving, you perform the same valid operation on both sides. Adding or subtracting the same number from both sides preserves equality. Multiplying or dividing both sides by the same nonzero number also preserves equivalence.

This balance image is useful because it reminds you that one side of an equation is not more important than the other. Whatever change you make to one side must be matched by an equivalent change on the other side.

Datei:Equation illustration.svg

Inverse Operations

Inverse operations undo each other.

Operation on the variable	Inverse operation used to solve	Example equation	Solution move
Add 7	Subtract 7	$x + 7 = 15$	Subtract 7 from both sides
Subtract 4	Add 4	$x - 4 = 9$	Add 4 to both sides
Multiply by 6	Divide by 6	$6 x = 42$	Divide both sides by 6
Divide by 5	Multiply by 5	$\frac{x}{5} = 3$	Multiply both sides by 5

A reliable question to ask is: What operation is being done to the variable, and what operation will undo it?

Solving Addition and Subtraction Equations

Addition Equations

Consider $x + 7 = 15$ . The variable has 7 added to it. Subtraction undoes addition, so subtract 7 from both sides:

$x + 7 - 7 = 15 - 7$

$x = 8$

Check the solution in the original equation: $8 + 7 = 15$ . The statement is true, so the solution is correct.

Another example is $x + (- 5) = 12$ . Adding negative 5 is the same as subtracting 5, so add 5 to both sides. The result is $x = 17$ .

Subtraction Equations

Consider $x - 9 = 4$ . The variable has 9 subtracted from it. Addition undoes subtraction, so add 9 to both sides:

$x - 9 + 9 = 4 + 9$

$x = 13$

A common mistake is to change only the left side. That breaks the balance. Always apply the inverse operation to both sides.

Datei:Equation illustration addition.svg

The illustration above shows the key idea: adding the same quantity to each side preserves a balanced equation.

Using a Number Line to Reason About Signed Values

Negative numbers often appear in one-step equations. A number line helps you interpret addition and subtraction as movement.

For $x + 4 = - 2$ , subtract 4 from both sides. Starting at -2 and moving 4 units left gives $x = - 6$ . For $x - 7 = - 10$ , add 7 to both sides, giving $x = - 3$ .

Solving Multiplication and Division Equations

Multiplication Equations

In $6 x = 42$ , the expression $6 x$ means 6 multiplied by $x$ . Division undoes multiplication, so divide both sides by 6:

$\frac{6 x}{6} = \frac{42}{6}$

$x = 7$

If the coefficient is negative, keep the sign with the coefficient. For $- 4 x = 20$ , divide both sides by -4, so $x = - 5$ .

The number multiplying a variable is called a coefficient. In $- 4 x = 20$ , the coefficient is -4.

Division Equations

In $\frac{x}{5} = - 3$ , the variable is divided by 5. Multiplication undoes division, so multiply both sides by 5:

$5 \cdot \frac{x}{5} = 5 \cdot (- 3)$

$x = - 15$

Check: $\frac{- 15}{5} = - 3$ . The equality is true.

Fractions and Decimals

The same inverse-operation idea works with fractions and decimals.

For $0.5 x = 7.5$ , divide both sides by 0.5. This gives $x = 15$ .

For $\frac{3}{4} x = 9$ , multiply both sides by the reciprocal $\frac{4}{3}$ . This gives $x = 12$ .

For $\frac{x}{\frac{2}{3}} = 6$ , multiply both sides by $\frac{2}{3}$ . This gives $x = 4$ .

A reciprocal is the number that multiplies another nonzero number to make 1. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$ .

A Four-Step Thinking Routine

Even though the equation itself needs only one inverse operation, a careful solver can use a short reasoning routine.

Identify the operation: Decide what operation is acting on the variable.
Choose the inverse operation: Select the operation that undoes it.
Preserve equality: Apply that operation to both sides and simplify.
Check by substitution: Put your result back into the original equation.

For $\frac{x}{- 3} = 4$ , the operation is division by -3. Its inverse is multiplication by -3. Multiplying both sides by -3 gives $x = - 12$ . Checking gives $\frac{- 12}{- 3} = 4$ , so the solution works.

Checking a Solution

Checking is part of solving, not an optional extra. Substitute the proposed value into the original equation and evaluate both sides.

Suppose you solved $5 x = - 35$ and got $x = - 7$ . Substitute -7:

Left side: $5 (- 7) = - 35$

Right side: $- 35$

Both sides have the same value, so the solution is correct.

If the two sides do not match, examine your inverse operation, arithmetic, and signs. A check can catch a small mistake before it becomes a larger one.

Translating Word Problems into One-Step Equations

An equation is a model of a relationship. To write one from a short situation, identify the unknown, decide what operation connects the known quantities, and write an equality.

Example 1: A game score was 12 points and then increased by an unknown number of points to reach 29. Let $p$ be the increase. The equation is $12 + p = 29$ . Subtract 12 from both sides, so $p = 17$ .

Example 2: Four identical notebooks cost 18 dollars in total. Let $n$ be the price of one notebook. The equation is $4 n = 18$ . Divide by 4, so $n = 4.5$ dollars.

Example 3: A temperature dropped 8 degrees and became -3 degrees. Let $t$ be the starting temperature. The equation is $t - 8 = - 3$ . Add 8 to both sides, so $t = 5$ degrees.

When writing equations, pay attention to units. A numerical answer without the correct interpretation may not fully answer the original question.

Common Errors and How to Fix Them

Changing only one side: If you subtract 7 on the left, you must subtract 7 on the right. Think of the equation as a balance.

Using the same operation instead of the inverse: If the equation is $x + 6 = 20$ , adding 6 again does not isolate the variable. Subtract 6.

Losing a negative sign: In $- 3 x = 18$ , divide by -3, not by 3. The solution is $x = - 6$ .

Confusing $x / 4$ with $4 / x$ : These are different expressions. In $x / 4 = 7$ , multiply both sides by 4.

Checking a changed equation instead of the original: Substitute your result into the equation you started with. That tests whether the value really solves the original problem.

Practice and Mathematical Communication

Strong algebra work includes both a correct result and a clear explanation. Instead of writing only $x = 8$ , you can say: "Seven is added to the variable, so I subtract 7 from both sides. This preserves equality and leaves $x = 8$ ."

Try to solve mentally when the numbers are easy, but still explain the inverse relationship. For more difficult fractions, decimals, or negative values, write the operation on both sides so your reasoning can be checked.

Interactive Tasks

Quiz: Test Your Knowledge

Which operation solves x plus 9 equals 14? (Subtract 9 from both sides) (!Add 9 to both sides) (!Multiply both sides by 9) (!Divide both sides by 9)

What is the solution of x minus 6 equals 11? (x equals 17) (!x equals 5) (!x equals negative 17) (!x equals negative 5)

What is the solution of 7x equals 49? (x equals 7) (!x equals 42) (!x equals 56) (!x equals 343)

Which inverse operation solves x divided by 4 equals 6? (Multiply both sides by 4) (!Divide both sides by 4) (!Add 4 to both sides) (!Subtract 4 from both sides)

What is the solution of negative 3x equals 18? (x equals negative 6) (!x equals 6) (!x equals negative 15) (!x equals 21)

What is the solution of x plus 5 equals negative 2? (x equals negative 7) (!x equals 3) (!x equals 7) (!x equals negative 3)

Why should you perform the same valid operation on both sides of an equation? (To preserve equality) (!To make both sides longer) (!To remove every constant) (!To change the variable name)

How do you check a proposed solution? (Substitute it into the original equation) (!Reverse the equals sign) (!Delete the variable) (!Round every number)

What is the solution of one half x equals 8? (x equals 16) (!x equals 4) (!x equals 8) (!x equals 32)

Which equation models four equal tickets costing 28 dollars in total? (Four times t equals 28) (!t plus 28 equals 4) (!t minus 4 equals 28) (!t divided by 28 equals 4)

Memory Game

Variable	A symbol that represents an unknown or changeable value
Equation	A statement that two expressions have equal value
Inverse operation	An operation that undoes another operation
Solution	A value that makes an equation true
Coefficient	A number multiplying a variable
Constant	A number whose value does not change
Substitution	Replacing a variable with a value to test or evaluate an expression

Drag and Drop

Match the correct terms.	Topic
Subtract seven	x plus seven equals fifteen
Add four	x minus four equals nine
Divide by six	six x equals forty two
Multiply by five	x divided by five equals three
Divide by negative three	negative three x equals eighteen

Match each inverse operation to the equation it solves.

Crossword Puzzle

Variable	What do you call a letter that represents an unknown value?
Inverse	What word describes an operation that undoes another operation?
Equality	What relationship does the equals sign state?
Solution	What do you call a value that makes an equation true?
Coefficient	What do you call the number multiplying a variable?
Substitute	What action means replacing a variable with a proposed value?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Equation Card Sort: Create eight cards with one-step equations and eight matching cards with the inverse operation needed to solve each equation, then test the sort with a partner.
Balance Drawing: Draw a balance-scale picture for one addition or subtraction equation and explain in two or three sentences why the same operation must be used on both sides.
Solution Check Recording: Choose four solved equations, substitute each proposed solution into the original equation, and record which checks are true.
Algebra Mini Poster: Make a one-page poster that shows the four inverse-operation pairs used in one-step equations and include one correct example for each pair.

Standard

Equation Interview: Interview a classmate about how they decide which inverse operation to use, then write a short summary comparing their strategy with yours.
Word Problem Builder: Write four original one-step word problems using different operations, solve them, and exchange them with another learner for checking.
Error Detective: Invent three believable incorrect solutions to one-step equations, identify the exact mistake in each, and write a corrected solution.
Equation Tutorial Video: Record a two-minute tutorial in which you solve one equation with a negative number and one equation with a decimal while explaining every step clearly.

Advanced

Fraction Equation Investigation: Create and solve six one-step equations involving fractions, including at least two with fractional coefficients, and explain how reciprocals help.
Real-World Equation Survey: Find at least three situations at home, school, a shop, sports practice, or another suitable place that can be modeled by one-step equations, then present the equations and interpretations.
Strategy Comparison Study: Compare mental solving, balance-model reasoning, and written inverse-operation solving on a set of eight equations, then argue when each strategy is most useful.
Equation Teaching Project: Design a short lesson for younger learners that includes a visual model, worked examples, a misconception warning, independent practice, and a method for checking solutions.

Learning Assessment

Explain the Balance Principle: Solve $x - 12 = - 5$ and justify each step by describing how equality is preserved.
Compare Two Methods: Solve $0.25 x = 6$ using division and again using a fraction or reciprocal idea, then explain why both methods produce the same result.
Diagnose an Error: A learner says that the solution of $- 5 x = 30$ is 6; identify the error, correct it, and verify the correction by substitution.
Model a Situation: A group pays 42 dollars for six equal-price tickets; define a variable, write a one-step equation, solve it, and interpret the answer with units.
Create and Justify: Write one addition, one subtraction, one multiplication, and one division equation that all have the solution $x = - 4$ , then explain how you know.
Transfer to a Formula: The formula $d = r t$ describes distance; if $d = 120$ and $r = 40$ , solve for $t$ and connect the procedure to one-step division equations.

Evidence of Learning

Evidence that you have learned this topic includes accurate knowledge of variables, constants, coefficients, equality, inverse operations, and solutions; the skill to solve one-step equations with integers, fractions, decimals, and negative values; clear written or spoken explanations of why an operation preserves equality; reliable checking by substitution; correct translation of short situations into equations; products such as posters, solution sets, videos, interviews, or mini-lessons; and transfer of the same reasoning to formulas, later multi-step equations, and related algebraic problems.

OERs on the Topic

For a broader mathematical context, the English Wikipedia article on linear equations explains equations in one or more variables and the meaning of their solutions.

Linked Learning Areas

Solving One-Step Equations

aiMOOC Projects