🔣 Solving Equations

Form and solve equations — one-step, two-step, and word problems.

One-step equation
x + 7 = 12 → x = 5
Two-step equation
3x + 4 = 19 → 3x = 15 → x = 5
Form from words
"I think of a number, double it, add 3 = 11" → 2n + 3 = 11

What you'll learn:

  • What an equation is (a balanced statement)
  • Solving one-step equations: + − × ÷
  • Solving two-step equations
  • Forming equations from word problems
  • Checking solutions by substituting back
Created by Mr Josh 🍭 FractionRush

📖 Solving Equations

What is an equation?

An equation has an equals sign and shows that two things balance. Our job is to find the value of the unknown letter.

Think of it like a balance scale: what's on the left equals what's on the right.
x + 5 = 12 means: "what number, plus 5, gives 12?"

Solving one-step equations

Do the inverse (opposite) operation to both sides to isolate the letter.

➕ Addition → subtract both sides
x + 7 = 15 → x = 15 − 7 = 8
➖ Subtraction → add both sides
x − 4 = 9 → x = 9 + 4 = 13
✖ Multiplication → divide both sides
5x = 30 → x = 30 ÷ 5 = 6
➗ Division → multiply both sides
x ÷ 3 = 7 → x = 7 × 3 = 21

Solving two-step equations

Work backwards — undo the last operation first, then the first operation.

Rule: Step 1 — deal with + or − first. Step 2 — deal with × or ÷.
Example: 2x + 3 = 11
Step 1: subtract 3 → 2x = 8
Step 2: divide by 2 → x = 4
Example: 4x − 5 = 19
Step 1: add 5 → 4x = 24
Step 2: divide by 4 → x = 6
Example: x/3 + 2 = 8
Step 1: subtract 2 → x/3 = 6
Step 2: multiply by 3 → x = 18

Forming equations from words

Read carefully and translate the words into algebra.

"I think of a number and add 6. The result is 20." → n + 6 = 20 → n = 14
"I think of a number, multiply by 3, then subtract 4. The answer is 11." → 3n − 4 = 11 → n = 5
"A rectangle has length (2x + 1) cm and width 4 cm. Its perimeter is 26 cm." → 2(2x + 1) + 2(4) = 26 → 4x + 2 + 8 = 26 → 4x = 16 → x = 4

Checking your answer

Always substitute your answer back to verify.

If 3x + 4 = 19 and you got x = 5, check: 3(5) + 4 = 15 + 4 = 19 ✓
If the two sides are equal, your answer is correct!

💡 Worked Examples

Example 1: One-step (addition)

Solve: x + 9 = 23

Subtract 9 from both sides: x = 23 − 9
x = 14
Check: 14 + 9 = 23 ✓
Example 2: One-step (multiplication)

Solve: 7n = 42

Divide both sides by 7: n = 42 ÷ 7
n = 6
Check: 7 × 6 = 42 ✓
Example 3: Two-step equation

Solve: 5x − 3 = 17

Step 1: Add 3 to both sides → 5x = 20
Step 2: Divide both sides by 5 → x = 4
x = 4
Check: 5(4) − 3 = 20 − 3 = 17 ✓
Example 4: Two-step (divide then add)

Solve: x/4 + 3 = 8

Step 1: Subtract 3 → x/4 = 5
Step 2: Multiply by 4 → x = 20
x = 20
Check: 20/4 + 3 = 5 + 3 = 8 ✓
Example 5: Forming an equation

"I think of a number. I multiply it by 4 and subtract 7. I get 21. Find the number."

Let the number = n
Equation: 4n − 7 = 21
Step 1: Add 7 → 4n = 28
Step 2: Divide by 4 → n = 7
The number is 7
Example 6: Perimeter problem

A rectangle has width 5 cm and length (3x − 2) cm. Its perimeter is 38 cm. Find x.

Perimeter = 2 × (length + width) = 38
2(3x − 2 + 5) = 38 → 2(3x + 3) = 38 → 6x + 6 = 38
6x = 32 → x = 32/6 ... wait, let's try: 6x + 6 = 38 → 6x = 32 → x = 16/3
Let's use a cleaner example: length = (2x + 3), width = 4, perimeter = 30
2(2x + 3 + 4) = 30 → 2(2x + 7) = 30 → 4x + 14 = 30 → 4x = 16 → x = 4

⚖️ Equation Balance Solver

Build a two-step equation and watch it solve step by step!

Build your equation: __

🎯 Drag 1: One-Step Addition

Solve each equation. Drag the correct answer into the box!

7
11
5
15
9
20
3
x + 4 = 11, so x = 
x + 6 = 17, so x = 
x + 9 = 14, so x = 
x + 8 = 23, so x = 
x + 7 = 16, so x = 

🎯 Drag 2: One-Step Multiplication

Divide to find the unknown!

6
8
4
9
7
5
10
5x = 30, so x = 
4x = 32, so x = 
7x = 28, so x = 
6x = 54, so x = 
3x = 21, so x = 

🎯 Drag 3: Subtract & Divide

Use inverse operations!

13
20
6
30
18
9
12
x − 5 = 8, so x = 
x − 7 = 13, so x = 
x ÷ 3 = 6, so x = 
x ÷ 5 = 6, so x = 
x − 3 = 9, so x = 

🎯 Drag 4: Two-Step Equations

Undo the + or − first, then the × or ÷!

4
5
3
7
6
9
2
2x + 3 = 11, so x = 
3x − 6 = 9, so x = 
4x + 5 = 17, so x = 
5x − 8 = 27, so x = 
3x + 12 = 30, so x = 

🎯 Drag 5: Form the Equation

Read the word problem and drag the correct equation!

2n+5=13
n−8=12
3n=24
n/4=6
4n−3=17
n+9=21
"I think of a number, double it, add 5, and get 13."
Equation: 
"I think of a number and subtract 8. The answer is 12."
Equation: 
"Three bags have the same number of sweets. Together there are 24."
Equation: 
"A number divided by 4 equals 6."
Equation: 

🎯 Drag 6: Mixed Challenge

One-step and two-step — mix of all types!

8
5
10
3
12
6
7
x + 13 = 21, so x = 
6x − 7 = 23, so x = 
x ÷ 4 + 2 = 4.5, so x = 
8x − 5 = 19, so x = 
x/2 + 6 = 12, so x = 

✏️ Practice Questions

20 questions — one-step, two-step, and forming equations.

  1. Solve: x + 8 = 17
  2. Solve: x − 5 = 12
  3. Solve: 4x = 36
  4. Solve: x ÷ 6 = 7
  5. Solve: 2x + 3 = 15
  6. Solve: 3x − 4 = 11
  7. Solve: 5x + 2 = 27
  8. Solve: x/3 + 5 = 9
  9. Solve: 7x − 6 = 29
  10. Solve: 2x + 9 = 25
  11. Solve: 4x − 7 = 13
  12. Solve: x/2 + 6 = 11
  13. "I think of a number, add 12, and get 30." Write an equation and solve it.
  14. "I multiply a number by 5 and get 45." Write an equation and solve it.
  15. "I think of a number, triple it, and subtract 9. The answer is 18." Write and solve.
  16. A rectangle has width 3 cm and length (2x + 1) cm. Its perimeter is 20 cm. Find x.
  17. The angles in a triangle are x°, 2x° and 60°. Find x. (Angles sum to 180°.)
  18. Solve: 6x − 11 = 37
  19. Solve: 3(x + 2) = 21
  20. "Five more than twice a number is 23." Form and solve the equation.
  1. x = 9
  2. x = 17
  3. x = 9
  4. x = 42
  5. x = 6
  6. x = 5
  7. x = 5
  8. x = 12
  9. x = 5
  10. x = 8
  11. x = 5
  12. x = 10
  13. n + 12 = 30 → n = 18
  14. 5n = 45 → n = 9
  15. 3n − 9 = 18 → 3n = 27 → n = 9
  16. 2(2x + 1) + 2(3) = 20 → 4x + 2 + 6 = 20 → 4x = 12 → x = 3
  17. x + 2x + 60 = 180 → 3x = 120 → x = 40°
  18. x = 8
  19. 3x + 6 = 21 → 3x = 15 → x = 5
  20. 2n + 5 = 23 → 2n = 18 → n = 9
Created by Mr Josh 🍭 FractionRush

🏆 Challenge: Word Problems

10 problems — form and solve equations from context.

  1. A bag of apples costs £x. Three bags cost £18. Write and solve an equation to find x.
  2. Mia thinks of a number, subtracts 14, and gets 9. What is her number?
  3. A plank of wood is (3x + 5) cm long. Another is 20 cm long. Together they measure 50 cm. Find x.
  4. The sum of three consecutive integers starting at n is 57. Find n. (Hint: n + (n+1) + (n+2) = 57)
  5. Josh is 4 years older than twice Ellie's age. Josh is 26. Write and solve an equation for Ellie's age.
  6. A rectangle has perimeter 44 cm. Its width is 8 cm. Let the length be x. Find x.
  7. "I think of a number. I add 7 to it, then multiply the result by 3. I get 36." Find the number.
  8. 6 pencils and 2 rulers cost £4.20. A ruler costs 60p more than a pencil. If a pencil costs p pence, write an equation and solve it. (All in pence.)
  9. Tickets for a show cost £(2x − 1) each. 5 tickets cost £40. Find x.
  10. A square has perimeter (8x + 12) cm and an actual perimeter of 60 cm. Find x and then find the side length.
  1. 3x = 18 → x = 6, so each bag costs £6
  2. n − 14 = 9 → n = 23
  3. (3x + 5) + 20 = 50 → 3x + 25 = 50 → 3x = 25 → x = 25/3 ≈ 8.33. (Cleaner: 3x + 5 = 30 → x = 25/3; or if the problem is set to give nice answer: try 50−20=30, 3x+5=30, 3x=25, x=25/3 ≈ 8.3 cm)
  4. n + n+1 + n+2 = 57 → 3n + 3 = 57 → 3n = 54 → n = 18
  5. 2e + 4 = 26 → 2e = 22 → e = 11. Ellie is 11.
  6. 2(x + 8) = 44 → x + 8 = 22 → x = 14 cm
  7. 3(n + 7) = 36 → n + 7 = 12 → n = 5
  8. Let p = price of pencil (pence). Ruler = p + 60. 6p + 2(p + 60) = 420 → 8p + 120 = 420 → 8p = 300 → p = 37.5p. Pencil costs 37.5p, ruler 97.5p.
  9. 5(2x − 1) = 40 → 2x − 1 = 8 → 2x = 9 → x = 4.5
  10. 8x + 12 = 60 → 8x = 48 → x = 6. Side length = 60 ÷ 4 = 15 cm.
Created by Mr Josh 🍭 FractionRush