Solve and verify multistep equations with integers and decimals

Ontario Grade 8 Mathematics

Algebra

Ontario Grade 8 Mathematics — C2.3

An equation is a statement that two expressions have the same value. Think of it as a balanced scale: if one side changes, the other side must change in the same way to stay balanced. A multistep equation takes more than one operation to solve. You can find the unknown number by undoing the operations that were done to it. Integers are positive and negative whole numbers, including zero. Decimals are numbers written with a decimal point. You will use both kinds of numbers in this lesson.

What you will learn

Explain that an equation shows two expressions with the same value.
Solve multistep equations with integers and decimals by keeping both sides balanced.
Use inverse operations in a helpful order.
Verify a solution by substituting it into the original equation.

Start with balance and inverse operations

A variable is a letter that stands for a number we do not know yet. In this lesson, the variable is often

x

. The equals sign means that the left side and the right side have the same value.

Imagine a balanced scale with an unknown amount on one side. If you add the same amount to each side, it stays balanced. Equations work in the same way. To keep an equation true, do the same operation to both sides.

Inverse operations undo each other. Addition and subtraction are inverse operations. Multiplication and division are also inverse operations. For example, subtracting

4

undoes adding

4

. Dividing by

3

undoes multiplying by

3

.

To solve an equation, undo the operations that affect the variable. Start with the operation farthest from the variable. In

4x+7=31

, the variable is multiplied by

4

, then

7

is added. Undo the addition first, then undo the multiplication.

a=b\implies a+c=b+c

An equation is balanced when both sides have the same value.
Do the same operation to both sides.
Undo operations in reverse order.

Follow a clear solving routine

First, look at each side of the equation. If a side has terms that can be combined, simplify it. Like terms have the same variable part. For example,

3x

and

2x

are like terms, but

3x

and

2

are not. Combine only terms that are alike.

Next, undo addition or subtraction connected to the variable term. Then undo multiplication or division. Write one balanced line at a time, and show what you do to both sides. This makes your work easier to follow and check.

Integers may have positive or negative signs. Keep each sign with its number as you calculate. For example, subtracting a negative number gives the same result as adding its positive value:

7-(-2)=7+2

.

For decimal addition and subtraction, line up the decimal points. When dividing decimals, calculate carefully and check the result in the original equation. The balance rule does not change just because the numbers are decimals.

Simplify like terms on one side before undoing operations.
Undo addition or subtraction before undoing multiplication or division.
Keep integer signs and decimal place value clear.

Verify your solution

A solution is a value for the variable that makes the original equation true. To verify means to check whether your answer works.

Substitute your answer for the variable in the original equation. Substitution means replacing the variable with the value you found. Then calculate the left side and the right side separately. If the two values match, your answer is verified.

For example, if you think

x=3

solves an equation, replace every

x

in the original equation with

3

. Do not check only the last line of your solving work. A mistake in an earlier step may still lead to a final line that looks reasonable.

If the values do not match, review your arithmetic and the operations you used on each side. Check signs carefully, especially when dividing by a negative number.

Substitute into the original equation.
Calculate both sides and compare their values.
Equal values confirm that the proposed solution works.

Practise the routine

Try these equations on paper:

3x-5=16

,

-2x+7=1

, and

1.5x+0.6=3.6

. For each one, write a balanced line for every operation. Then substitute your answer into the original equation.

Use the same routine each time: identify the operations, undo them in reverse order, and verify. You may be solving with positive integers, negative integers, or decimals, but the goal stays the same: find a value that makes both sides equal.

Show each operation clearly.
Check every answer in the original equation.
A correct check has equal values on both sides.

Worked example

A positive integer equation

Solve and verify

4x+7=31

.

Undo the addition
The variable term has $7$ added to it. Subtract $7$ from both sides to keep the equation balanced.
$4x+7-7=31-7$
Undo the multiplication
Now $4$ is multiplied by $x$ . Divide both sides by $4$ to find the value of $x$ .
$\frac{4x}{4}=\frac{24}{4}$
Verify in the original equation
Replace $x$ with $6$ in the original equation. Both sides equal $31$ , so the solution works.
$4(6)+7=31$

Answer:

x=6

Check: The left side is

4(6)+7=31

. The right side is

31

, so the values match.

Worked example

An equation with a negative coefficient

Solve and verify

-3x-5=10

.

Undo subtraction
The left side has $5$ subtracted. Add $5$ to both sides to keep them equal.
$-3x-5+5=10+5$
Undo multiplication
Now $-3$ is multiplied by $x$ . Divide both sides by $-3$ . A positive number divided by a negative number gives a negative result.
$\frac{-3x}{-3}=\frac{15}{-3}$
Verify the value
Substitute $-5$ into the original equation. The left side becomes $10$ , equal to the right side.
$-3(-5)-5=10$

Answer:

x=-5

Check: The left side is

-3(-5)-5=15-5=10

. This matches the right side.

Worked example

A decimal equation

Solve and verify

2.5x+1.2=6.2

.

Undo the addition
Subtract $1.2$ from both sides. The decimal subtraction gives $5.0$ on the right.
$2.5x+1.2-1.2=6.2-1.2$
Undo multiplication
Divide both sides by $2.5$ . This leaves the variable by itself.
$\frac{2.5x}{2.5}=\frac{5.0}{2.5}$
Verify
Replace $x$ with $2$ in the original equation. The left side equals $6.2$ , so the solution works.
$2.5(2)+1.2=6.2$

Answer:

x=2

Check: The left side is

2.5(2)+1.2=5.0+1.2=6.2

. This equals the right side.

Common mistakes and how to avoid them

Subtracting or dividing on only one side.

Correction: Apply the same operation to both sides so the equation stays balanced.

Undoing multiplication before undoing an added or subtracted number.

Correction: Undo the operation farthest from the variable first. In

4x+7

, subtract

7

before dividing by

4

.

Losing a negative sign while dividing.

Correction: Keep the sign in the division. For example,

15\div(-3)=-5

.

Checking only the final solving line.

Correction: Substitute your answer into the original equation and compare both sides.

Lesson summary

An equation is balanced when both sides have equal values.
Use the same operation on both sides.
Undo operations in reverse order.
Watch integer signs and decimal place value.
Verify by substituting the answer into the original equation.

Check your understanding

Question 1

Solve

5x-4=21

. Which value of

x

makes the equation true?

$3$
$5$
$25$
$17$

Show answer and explanation

5

Add

4

to both sides to get

5x=25

, then divide by

5

. Checking gives

5(5)-4=21

.

Question 2

Solve

-2x+3=11

. What is

x

?

$-4$
$4$
$7$
$-7$

Show answer and explanation

-4

Subtract

3

from both sides to get

-2x=8

. Divide by

-2

to get

x=-4

. Substitution gives

-2(-4)+3=11

.

Question 3

Solve

1.2x+0.4=2.8

. What is

x

?

$1$
$2$
$2.4$
$3$

Show answer and explanation

2

Subtract

0.4

from both sides to get

1.2x=2.4

. Divide by

1.2

to get

x=2

. Checking gives

1.2(2)+0.4=2.8

.

Key terms

Equation: A statement that two expressions have the same value.
Variable: A letter that stands for a number that may be unknown.
Inverse operations: Pairs of operations that undo each other, such as addition and subtraction.
Solution: A value for the variable that makes the equation true.
Verify: Check a solution by substituting it into the original equation and comparing both sides.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation C2.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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C2.3 · Solve and verify multistep equations with integers and decimals

Ontario Grade 8 Mathematics

Ontario Grade 8 Mathematics — C2.3

What you will learn

Start with balance and inverse operations

Follow a clear solving routine

Verify your solution

Practise the routine

A positive integer equation

An equation with a negative coefficient

A decimal equation

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review