Create, solve, and verify equations

Ontario Grade 9 Mathematics

Algebra

Ontario MTH1W — Expectation C1.5

Every time you split a restaurant bill, figure out how many hours you need to work to afford something, or measure ingredients for a recipe, you are doing the same thinking that algebra formalises. An equation is a mathematical sentence that says two expressions are equal. Learning to write, solve, and check equations gives you a reliable method for answering "what is the unknown?" in almost any number situation. This lesson walks you through each stage: turning a word problem into an equation, solving it step by step, and then confirming your answer is correct.

What you will learn

Translate a real-life situation into an equation using a variable.
Solve a linear equation by applying inverse operations in the correct order.
Verify a solution by substituting it back into the original equation.
Identify and fix common equation-building and solving errors.

What Is an Equation? A Concrete Starting Point

Imagine you and a friend buy matching hoodies. You each pay the same price. Together you spend CAD 90. The question is: how much does one hoodie cost? You already know the answer is CAD 45, because CAD 45 + CAD 45 = CAD 90. That mental step is exactly what an equation captures in symbols.

An equation is a statement that two expressions have the same value, connected by an equals sign. The equals sign (

=

) is a balance point — whatever is on the left must weigh the same as whatever is on the right. If you change one side, you must change the other side in the same way to keep the balance.

A variable is a letter that stands for an unknown number. In the hoodie situation, let

h

represent the cost of one hoodie in dollars. Because two hoodies cost CAD 90, the equation is

2h = 90

. The variable

h

is the quantity you do not yet know but are trying to find.

\text{left expression} = \text{right expression}

An equation states that two expressions are equal.
The equals sign works like a balance: both sides must stay equal.
A variable is a letter that represents an unknown number.
One equation can have exactly one unknown when there is one variable.

Building an Equation from a Situation

Before you can solve a problem, you must translate it into an equation. This is called creating or setting up the equation. There are four steps to follow every time.

Step 1 — Identify the unknown. Decide what quantity you do not know and give it a variable name. Choose a letter that reminds you of the quantity, for example

n

for a number of items or

t

for time. Step 2 — Find the relationships. Read the problem carefully and list what operations connect the unknown to the numbers you do know. Words like 'more than' suggest addition, 'times as many' suggests multiplication, 'shared equally' suggests division, and 'less than' suggests subtraction. Step 3 — Write the equation. Translate each part of the sentence into a mathematical expression and join them with an equals sign. Step 4 — Check that the equation matches the story before you solve it.

Consider this situation: a streaming service charges a flat monthly fee of CAD 8, plus CAD 3 for every movie rented. One month, a customer's bill is CAD 20. How many movies were rented? Let

m

stand for the number of movies. The flat fee is CAD 8. Each movie adds CAD 3, so

m

movies add

3m

dollars in total. The total bill is CAD 20. The equation is

3m + 8 = 20

. Notice how each part of the story appears in the equation.

3m + 8 = 20

Identify the unknown first and name it with a variable.
Look for key words that signal which operation to use.
Write the equation so every piece of information in the problem is represented.
Re-read the problem to confirm the equation matches before solving.

Solving an Equation Using Inverse Operations

To solve an equation means to find the value of the variable that makes the equation true. The main tool is inverse operations. An inverse operation is the opposite of a given operation: addition and subtraction are inverses of each other, and multiplication and division are inverses of each other.

The goal is to isolate the variable — to get it alone on one side of the equals sign. You do this by performing the same inverse operation on both sides so the balance is not disturbed. Work in reverse order: undo addition or subtraction first (these are the outside operations), then undo multiplication or division (these are closer to the variable). This is the opposite of the order of operations you learned in Grades 7 and 8.

Using the streaming example, where the equation is

3m + 8 = 20

: first, undo the addition of 8 by subtracting 8 from both sides, giving

3m = 12

. Next, undo the multiplication by 3 by dividing both sides by 3, giving

m = 4

. The solution is

m = 4

, meaning 4 movies were rented.

Here is a useful way to think about it: picture the equation as a locked box. Each operation on the variable is a lock. To open the box, you remove the locks in the reverse order they were put on, doing the same thing to both sides every time.

3m + 8 - 8 = 20 - 8 \implies 3m = 12 \implies m = 4

Isolating the variable means getting it alone on one side.
Use inverse operations: subtraction undoes addition; division undoes multiplication.
Apply every operation to both sides to keep the equation balanced.
Undo operations in the reverse order they appear (outside operations first).

Verifying the Solution

Verification means checking that your answer is correct. Never skip this step — it takes only a few seconds and will catch arithmetic errors before they cost you marks.

To verify, substitute your solution back into the original equation. Replace the variable with the value you found. Then simplify both sides separately and confirm they are equal. If they are equal, the solution is correct. If they are not, you made an error somewhere and need to re-solve.

Verifying

m = 4

in the original equation

3m + 8 = 20

: substitute to get

3(4) + 8

. Calculate the left side:

3 \times 4 = 12

, then

12 + 8 = 20

. The right side is also

20

. Since both sides equal

20

, the solution

m = 4

is verified.

Verification is not just a checking habit — it also helps you understand the problem. When you substitute and the numbers work out, you can see exactly why the equation was set up the way it was.

3(4) + 8 = 12 + 8 = 20 \checkmark

Substitute your answer back into the original equation to verify.
Simplify both sides separately, then compare.
If both sides are equal, the solution is correct.
Always verify using the original equation, not a rearranged version.

Equations with Variables on Both Sides

Sometimes a situation leads to an equation where the variable appears on both sides of the equals sign. You need one extra step: collect all the variable terms on one side before isolating the variable.

Here is a situation that produces this type of equation. Two friends, Priya and Jordan, are saving money. Priya already has CAD 15 and saves CAD 10 per week. Jordan has CAD 5 and saves CAD 12 per week. After how many weeks will they have the same amount? Let

w

represent the number of weeks. Priya's total is

10w + 15

dollars. Jordan's total is

12w + 5

dollars. Setting these equal gives the equation

10w + 15 = 12w + 5

.

To solve, first move all variable terms to one side. Subtract

10w

from both sides to get

15 = 2w + 5

. Next, subtract 5 from both sides to get

10 = 2w

. Finally, divide both sides by 2 to get

w = 5

. After 5 weeks, both friends will have the same amount. Verify: Priya has

10(5) + 15 = 65

dollars and Jordan has

12(5) + 5 = 65

dollars, so

w = 5

is correct.

10w + 15 = 12w + 5 \implies 10 = 2w \implies w = 5

If the variable appears on both sides, collect variable terms on one side first.
Subtract the smaller variable term from both sides to keep the coefficient positive.
Then continue isolating the variable as usual.
Always verify using both expressions from the original problem.

Key Words and the Operations They Signal

Key word or phrase	Operation it usually signals	Example
more than, increased by, total	Addition ( $+$ )	'5 more than $n$ ' → $n + 5$
less than, decreased by, difference	Subtraction ( $-$ )	'7 less than $n$ ' → $n - 7$
times, product of, twice, double	Multiplication ( $\times$ )	'Twice $n$ ' → $2n$
shared equally, split into, quotient	Division ( $\div$ )	' $n$ split into 4' → $n \div 4$
is, equals, the result is, gives	Equals sign ( $=$ )	'The total is 30' → $= 30$

Worked example

A Perimeter Problem: Create, Solve, and Verify

A rectangular garden has a length that is 4 m more than twice its width. The perimeter of the garden is 50 m. Find the width and the length of the garden.

Step 1: Identify the unknown
The width is the quantity we know least about, so let $w$ represent the width in metres. The length is described in terms of the width, so we can write it using $w$ as well.
$w = \text{width (m)}$
Step 2: Write an expression for the length
'4 m more than twice the width' means multiply the width by 2 and then add 4.
$\text{length} = 2w + 4$
Step 3: Create the equation
The perimeter of a rectangle equals two lengths plus two widths. Set this expression equal to 50.
$2(2w + 4) + 2w = 50$
Step 4: Expand the brackets
Multiply each term inside the brackets by 2. Recall that $2 \times 2w = 4w$ and $2 \times 4 = 8$ .
$4w + 8 + 2w = 50$
Step 5: Collect like terms on the left side
Combine the two variable terms: $4w + 2w = 6w$ . The equation becomes simpler.
$6w + 8 = 50$
Step 6: Subtract 8 from both sides
Undo the addition of 8 by subtracting 8 from both sides to isolate the variable term.
$6w = 42$
Step 7: Divide both sides by 6
Undo the multiplication by 6 by dividing both sides by 6.
$w = 7$
Step 8: Find the length
Substitute $w = 7$ into the length expression $2w + 4$ .
$\text{length} = 2(7) + 4 = 14 + 4 = 18$
Step 9: Verify
Calculate the perimeter using width = 7 m and length = 18 m and check it equals 50 m.
$2(18) + 2(7) = 36 + 14 = 50 \checkmark$

Answer: The width is 7 m and the length is 18 m.

Check: Perimeter

= 2(18) + 2(7) = 36 + 14 = 50

m. This matches the given perimeter, so the solution is correct.

Common mistakes and how to avoid them

Forgetting to apply an operation to both sides. For example, subtracting a number from only the left side and leaving the right side unchanged.

Correction: Every operation you perform must be done to both sides of the equation at the same time. Think of the equals sign as a balance: one side cannot change without the other.

Undoing operations in the wrong order. For example, dividing before subtracting when the equation is

3m + 8 = 20

, which gives

m + 8 \approx 6.67

instead of correctly subtracting 8 first.

Correction: Undo operations from the outside in — deal with addition or subtraction that is separate from the variable term first, then deal with multiplication or division attached to the variable.

Setting up the equation incorrectly by reversing a phrase. For example, translating '4 more than twice the width' as

4(2w)

instead of

2w + 4

.

Correction: Break the phrase into small pieces. 'Twice the width' is

2w

. '4 more than that' means add 4, giving

2w + 4

. Re-read the phrase against your expression before moving on.

Verifying with a rearranged or simplified version of the equation rather than the original. This can hide errors made during set-up.

Correction: Always substitute your answer into the original equation exactly as it was written before any solving steps. This is the only reliable check.

Dropping a negative sign when moving terms across the equals sign. For example, writing

10w + 15 = 12w + 5

and then incorrectly getting

15 = 2w - 5

instead of

15 = 2w + 5

.

Correction: When you subtract a term from both sides, apply the subtraction carefully. Rewrite the full equation after each step and check the signs before continuing.

Lesson summary

An equation states that two expressions are equal, connected by an equals sign that acts like a balance.
To create an equation: name the unknown with a variable, identify the relationships, and translate the situation into mathematical expressions joined by $=$ .
To solve an equation: use inverse operations (opposite operations) to isolate the variable, applying every step to both sides.
When the variable appears on both sides, collect all variable terms on one side first, then isolate the variable.
To verify a solution: substitute the answer back into the original equation, simplify both sides, and confirm they are equal.
Use a table of key words to translate between plain language and mathematical operations when building equations.

Check your understanding

Question 1

A phone plan costs CAD 25 per month plus CAD 0.10 per text message. One month the bill is CAD 31. Which equation correctly represents this situation, where

t

is the number of text messages?

$0.10t = 31$
$25t + 0.10 = 31$
$0.10t + 25 = 31$
$25t - 0.10 = 31$

Show answer and explanation

0.10t + 25 = 31

The fixed monthly cost is CAD 25. Each text costs CAD 0.10, so

t

texts cost

0.10t

dollars in total. Adding the fixed cost gives the total bill:

0.10t + 25 = 31

.

Question 2

Solve

5x - 7 = 18

. What is the value of

x

?

$x = 2.2$
$x = 5$
$x = 11$
$x = 25$

Show answer and explanation

x = 5

Add 7 to both sides:

5x = 25

. Divide both sides by 5:

x = 5

. Verify:

5(5) - 7 = 25 - 7 = 18

✓

Question 3

A student solved

4n + 3 = 23

and got

n = 6.5

. Which step shows what went wrong?

The student divided by 4 before subtracting 3, giving $n + 3 = 5.75$ .
The student subtracted 3 from both sides first, giving $4n = 20$ , then divided to get $n = 5$ .
The student added 3 to both sides, giving $4n = 26$ , then divided to get $n = 6.5$ .
The student multiplied both sides by 4 first.

Show answer and explanation

The student added 3 to both sides, giving

4n = 26

, then divided to get

n = 6.5

.

The error was adding 3 to both sides instead of subtracting 3. The correct step is

4n + 3 - 3 = 23 - 3

, giving

4n = 20

, then dividing by 4 gives

n = 5

. Verify:

4(5) + 3 = 23

✓

Question 4

Two classmates are collecting canned food. Marcus has 10 cans and collects 6 more per day. Leila has 22 cans and collects 2 more per day. After how many days will they have the same number of cans?

2 days
3 days
4 days
5 days

Show answer and explanation

3 days

Let

d

= number of days. Marcus:

6d + 10

. Leila:

2d + 22

. Set equal:

6d + 10 = 2d + 22

. Subtract

2d

from both sides:

4d + 10 = 22

. Subtract 10:

4d = 12

. Divide by 4:

d = 3

. Verify: Marcus has

6(3) + 10 = 28

; Leila has

2(3) + 22 = 28

✓

Key terms

Equation: A mathematical statement that says two expressions are equal, written with an equals sign between them.
Variable: A letter used to represent an unknown number in an equation or expression.
Expression: A combination of numbers, variables, and operations that represents a quantity but does not include an equals sign.
Inverse operation: An operation that reverses the effect of another operation. Addition and subtraction are inverses; multiplication and division are inverses.
Isolate the variable: To rearrange an equation so the variable is alone on one side of the equals sign, revealing its value.
Verify: To check a solution by substituting it back into the original equation and confirming both sides are equal.
Substitution: Replacing a variable with a specific number to evaluate an expression or check a solution.
Like terms: Terms in an expression that have the same variable part. For example, $3x$ and $5x$ are like terms and can be combined into $8x$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation C1.5. 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.

Official curriculum reference

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C1.5 · Create, solve, and verify equations

Ontario Grade 9 Mathematics

Ontario MTH1W — Expectation C1.5

What you will learn

What Is an Equation? A Concrete Starting Point

Building an Equation from a Situation

Solving an Equation Using Inverse Operations

Verifying the Solution

Equations with Variables on Both Sides

Key Words and the Operations They Signal

A Perimeter Problem: Create, Solve, and Verify

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review