Equations — Solve Linear Equations up to 2 Steps

Before you begin

Key Words

Click each card to flip it and see the full definition, examples, and non-examples. Make sure you understand all 6 words before moving on.

Pronumeral

A letter that stands for a number

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Definition

A letter used to represent an unknown or changing number.

Examples

x, y, n, a, b
"Let x = the unknown"

Not Examples

The number 5
A word like "seven"

Key idea

It can take any value until we solve

In a sentence

"Let n be the number of apples."

Expression

Numbers, letters and operations

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Definition

A combination of numbers, pronumerals and operations. Has NO equals sign.

Examples

3x + 5
2n − 1
4(y + 2)

Not Examples

3x + 5 = 11 (this is an equation!)

Think of it as

A mathematical phrase (no "is equal to")

Key rule

No = sign. Cannot be "solved".

Equation

Two things that are equal

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Definition

A mathematical statement showing two expressions are equal. Always has an = sign.

Examples

3x + 5 = 11
2n = 14
x − 3 = 7

Not Examples

3x + 5 (no equals sign — expression)

Key feature

Must have an = sign

Can be

Solved to find the value of the pronumeral

Solution

The value that makes it true

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Definition

The value of the pronumeral that makes the equation true.

Example

x + 3 = 7
Solution: x = 4

Check it

Substitute back: 4 + 3 = 7 ✓

Also called

The "root" of the equation

Key idea

Always check by substituting

Inverse Operation

The "undo" operation

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Definition

The operation that reverses or "undoes" another operation.

Pairs

+ undoes −
× undoes ÷
− undoes +

Example

x + 5 = 12
Subtract 5 from both sides

Think of it as

Going backwards to "unwrap" x

Key rule

Do the same thing to BOTH sides

Coefficient

The number in front of a pronumeral

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Definition

The number multiplied by a pronumeral in an expression or equation.

Examples

In 3x: coefficient is 3
In 7n: coefficient is 7

Special case

x means 1x — coefficient is 1 (not written)

Why it matters

To solve 3x = 12, divide both sides by 3

Not the coefficient

The + 5 in 3x + 5 (that's a constant)

✅ Remember: In algebra, 3x means "3 multiplied by x", and x4 means "x divided by 4". The pronumeral x can represent any number — our job when solving is to find out which number it is.

Core concept

Expressions vs Equations

This is the most important distinction in this topic. Make sure you understand the difference before solving.

Expression

3x + 5

A mathematical phrase. Has NO equals sign. Cannot be solved — only simplified.

2n − 1
4(y + 2)
x² + 3x

Equation

3x + 5 = 11

A mathematical sentence. HAS an equals sign. Can be solved to find the unknown.

2n − 1 = 7
4(y + 2) = 20
x + 3 = x − 1 ✗

💡 The equals sign is a balance point. It means "the left side has the same value as the right side." An equation is like a set of scales — both sides must always stay balanced.

Writing equations from number sentences

When a number sentence has an unknown quantity, we use a pronumeral to represent it.

Word sentence

A number plus 7 equals 12

Equation

n + 7 = 12

Word sentence

Three times a number equals 18

Equation

3n = 18

Word sentence

Double a number, subtract 3, gives 11

Equation

2n − 3 = 11

🧠 Quick Check

Which of the following is an equation?

Visual model

The Balance Scale Model

An equation is like a balance scale — both sides must always be equal. Whatever you do to one side, you must do to the other.

2x + 3 = 11

The scale is balanced. Both sides equal 11 when x = 4.

Try doing operations to both sides. Watch what happens!

⚖️ The Golden Rules of Equation Solving

Rule 1 Whatever you do to one side, do the same to the other side.

Rule 2 Use inverse operations to "undo" what's been done to x.
+ undoes −, × undoes ÷

Rule 3 Aim to get x by itself (isolate the pronumeral).

Rule 4 Always check: substitute your answer back in.

🗝️ How to "unwrap" x in a 2-step equation:

Think of the equation as a wrapped parcel. To open it, you undo the operations in reverse order:

2x + 3 = 11
Step 1: subtract 3 (undo +3)
Step 2: divide by 2 (undo ×2)

x3 − 4 = 2
Step 1: add 4 (undo −4)
Step 2: multiply by 3 (undo ÷3)

I Do

Worked Examples

Read each example carefully. Click Show Next Step to work through the solution one step at a time.

Example 1 — One-step equation (subtraction)

Foundation

x + 7 = 15

→

The equation:

x + 7 = 15

We need to find the value of x. x has 7 added to it.

Inverse operation: subtract 7 from both sides

x + 7 − 7 = 15 − 7

We undo the "+7" by subtracting 7. Do this to BOTH sides.

Simplify

x = 8

+7 − 7 cancels out, leaving x by itself.

✓

Check: substitute x = 8 back into the original equation

8 + 7 = 15 ✓

Both sides equal 15 — the answer is correct!

Example 2 — One-step equation (division)

Foundation

4x = 28

→

The equation:

4x = 28

4x means 4 × x. We need to undo the multiplication by 4.

Inverse operation: divide both sides by 4

4x ÷ 4 = 28 ÷ 4

We undo "×4" by dividing by 4. Do this to BOTH sides.

Simplify

x = 7

4 ÷ 4 = 1, so 4x ÷ 4 = x. And 28 ÷ 4 = 7.

✓

Check: substitute x = 7

4 × 7 = 28 ✓

Correct!

Example 3 — Two-step equation

Core

3x − 4 = 11

→

The equation:

3x − 4 = 11

Two things have been done to x: multiplied by 3, then 4 subtracted. We undo in reverse order.

Undo the −4 first: add 4 to both sides

3x − 4 + 4 = 11 + 4

We deal with the constant (number without x) first.

Simplify

3x = 15

−4 + 4 = 0, so the left side simplifies to 3x. Right side: 11 + 4 = 15.

Undo the ×3: divide both sides by 3

3x ÷ 3 = 15 ÷ 3

Dividing by 3 undoes the multiplication.

Solution

x = 5

✓

Check: substitute x = 5

3(5) − 4 = 15 − 4 = 11 ✓

Example 4 — Pronumerals on both sides

Core / Extension

5x + 2 = 2x + 14

→

The equation:

5x + 2 = 2x + 14

x appears on both sides. First, collect all the x terms on one side.

Subtract 2x from both sides

5x − 2x + 2 = 2x − 2x + 14

This removes the x term from the right side.

Simplify

3x + 2 = 14

5x − 2x = 3x. The right side: 2x − 2x = 0, leaving 14.

Subtract 2 from both sides

3x = 12

2 − 2 = 0 on the left. 14 − 2 = 12 on the right.

Divide both sides by 3

x = 4

✓

Check: substitute x = 4 into both sides

Left: 5(4)+2 = 22 Right: 2(4)+14 = 22 ✓

Example 5 — Non-integer solution (fraction answer)

Extension

2x + 1 = 6

→

The equation:

2x + 1 = 6

The answer will not be a whole number — that's okay!

Subtract 1 from both sides

2x = 5

Divide both sides by 2

x = 52 = 2.5

5 ÷ 2 = 2.5 or 2½. Both forms are correct.

✓

Check: substitute x = 2.5

2(2.5) + 1 = 5 + 1 = 6 ✓

Spot the mistake

Error Detective

Each solution below contains exactly ONE mistake. Find which line is wrong, then click "Reveal the Error" to check your thinking.

🔍

Detective Case 1 — Find the error in solving: 4x − 2 = 10

Read each line. One line has a mistake. Which line is it?

Line 1: 4x − 2 = 10

Line 2: 4x − 2 + 2 = 10 + 2

Line 3: 4x = 12

Line 4: x = 12 − 4 → x = 8

Line 5: Check: 4(8) − 2 = 30 ≠ 10 ✗

I think the error is on line:

🔍 The error is on Line 4.

The equation at Line 3 is 4x = 12. To undo ×4, you must divide both sides by 4 — not subtract.

Correct Line 4: 4x ÷ 4 = 12 ÷ 4 → x = 3

Check: 4(3) − 2 = 12 − 2 = 10 ✓

Common mistake: students subtract the coefficient instead of dividing by it.

🔍

Detective Case 2 — Find the error in solving: 3x + 5 = 2x + 9

Line 1: 3x + 5 = 2x + 9

Line 2: 3x − 2x + 5 = 2x − 2x + 9 → x + 5 = 9

Line 3: x + 5 − 5 = 9 + 5

Line 4: x = 14

Line 5: Check: 3(14)+5 = 47 ≠ 2(14)+9 = 37 ✗

I think the error is on line:

🔍 The error is on Line 3.

At Line 2 we correctly reach x + 5 = 9. To undo "+5", we must subtract 5 from both sides. But Line 3 subtracts 5 from the left and adds 5 to the right — breaking the balance.

Correct Line 3: x + 5 − 5 = 9 − 5 → x = 4

Check: 3(4)+5 = 17 2(4)+9 = 17 ✓

Common mistake: not applying the same operation to both sides.

Modelling with equations

Word Problems

Word problems ask you to translate a real situation into an equation, then solve it. Use the 5-step framework below for every problem.

📋 The 5-Step Framework

1. Read
Read the problem. Identify the unknown.

2. Define
Let x = the unknown. State this clearly.

3. Write
Form an equation from the information.

4. Solve
Solve the equation using algebra.

5. Answer
Write a sentence. Check it makes sense.

Problem 1 — One-step

"A student thinks of a number. They add 9 to it and the result is 22. What is the number?"

1 What is the unknown? Define it with a pronumeral.

Let = the number I am thinking of.

2 Write the equation.

The number plus 9 equals 22, so the equation is:

3 Solve and check.

I used the inverse operation () to find x = . I checked by substituting: ✓

✅ Full Solution

Let n = the unknown number

Equation: n + 9 = 22

Solve: n + 9 − 9 = 22 − 9 ∴ n = 13

Check: 13 + 9 = 22 ✓

Answer: The number is 13.

Problem 2 — Two-step

"Tickets to a school play cost $5 each. Priya buys some tickets and also pays a $3 booking fee. She spends $23 in total. How many tickets did she buy?"

1 What is the unknown?

Let = the number of .

2 Write the equation. (Think: cost of tickets + booking fee = total)

= 23

3 Solve and write your answer as a sentence.

Priya bought tickets.

✅ Full Solution

Let t = number of tickets

Equation: 5t + 3 = 23

Step 1: 5t + 3 − 3 = 23 − 3 → 5t = 20

Step 2: 5t ÷ 5 = 20 ÷ 5 → t = 4

Check: 5(4) + 3 = 20 + 3 = 23 ✓

Answer: Priya bought 4 tickets.

Problem 3 — Two groups (pronumerals both sides)

"Marcus and Jordan are saving money. Marcus has saved $6 more than three times what Jordan has. Jordan has saved twice what Marcus started with. If Marcus has $x and Jordan also has $x, and Marcus's total is 3x + 6 while Jordan's total is 2x + 10. They have the same amount. Find x."

1 Write the equation (both totals are equal).

3x + 6 =

2 Collect x terms on one side, then solve.

x =

✅ Full Solution

Equation: 3x + 6 = 2x + 10

Step 1: 3x − 2x + 6 = 2x − 2x + 10 → x + 6 = 10

Step 2: x + 6 − 6 = 10 − 6 → x = 4

Check: 3(4)+6 = 18 2(4)+10 = 18 ✓

Answer: x = 4

You Do

Practice Questions

Choose your level. Each question has a hint if you get stuck, and feedback that explains why.

Foundation: One-step equations and identifying expressions vs equations. Take your time — use inverse operations!

Q1. Which of these is an equation?

💡 Need a hint?

An equation must have an equals sign (=). Look for the option that has one.

Q2. Solve: x + 9 = 16

💡 Need a hint?

The inverse of adding 9 is subtracting 9. Subtract 9 from both sides.

Q3. Solve: 5x = 35

💡 Need a hint?

5x means 5 × x. To undo multiplication, divide both sides by 5.

Q4. Solve: x − 4 = 11

💡 Need a hint?

The inverse of − 4 is + 4. Add 4 to both sides.

Core: Two-step equations. Remember: undo the constant first, then undo the coefficient.

Q1. Solve: 2x + 3 = 13

💡 Need a hint?

Two-step: (1) subtract 3 from both sides → 2x = 10. (2) divide by 2.

Q2. Solve: 4x − 7 = 17

💡 Need a hint?

Step 1: Add 7 to both sides → 4x = 24. Step 2: Divide by 4.

Q3. Solve: x3 + 2 = 7

💡 Need a hint?

Step 1: Subtract 2 → x/3 = 5. The inverse of dividing by 3 is multiplying by 3.

Q4. Sam earns $d per hour and works for 3 hours. He also earns a $5 bonus. His total pay is $26. Which equation models this?

💡 Need a hint?

Total pay = (hourly rate × hours) + bonus. Translate each part into algebra.

Extension: Pronumerals on both sides, non-integer solutions, and more complex word problems.

Q1. Solve: 5x + 1 = 3x + 9

💡 Need a hint?

First: collect x terms on one side by subtracting 3x. Then solve the resulting 2-step equation.

Q2. Solve: 3x + 4 = x + 9. What is x?

💡 Need a hint?

Subtract x from both sides first. You'll get 2x + 4 = 9. Then solve the two-step equation.

Q3. Solve: 2(x + 3) = 14

💡 Need a hint?

The bracket means the whole bracket is multiplied by 2. Divide both sides by 2 first to "remove" the factor.

Q4. Jordan is 3 years older than twice Marcus's age. If Jordan's age is also equal to Marcus's age plus 11, how old is Marcus? (Let m = Marcus's age.)

💡 Need a hint?

Jordan's age = 2m + 3 AND Jordan's age = m + 11. So 2m + 3 = m + 11. Solve for m.