Solving Algebraic Equations
Learn to find the value of unknowns using equations
1. What is an Equation?
The Big Idea
An equation is a statement that two things are equal. It has an equals sign (=).
Solving an equation means finding the value of the unknown that makes the equation true.
Expression vs Equation
Expression
No equals sign
3x + 5
2y - 7
Equation
Has equals sign
3x + 5 = 20
2y - 7 = 11
Thinking About Equations
An equation is like a balanced scale - both sides must be equal!
Key Insight: In an equation like 9g = 72, finding g is like finding what 1 unit equals!
Think of it as finding the value of 1 unit in bar model problems.
2. One-Step Equations
One-step equations can be solved in a single calculation.
Example 1: Siti's Crayons
Siti had 7 crayons. She then bought j more crayons. She had 18 crayons in the end. How many crayons did Siti buy?
Setting up the equation:
7 + j = 18
Solving:
j = 18 - 7
j = 11
Siti bought 11 crayons.
Example 2: Solving 9g = 72
Find the value of g if 9g = 72.
9 units = 72, so 1 unit = ?
9g = 72
g = 72 ÷ 9
g = 8
Practice: One-Step Equation
Solve: 3h = 12. Find the value of h.
ShowHide solution
3h = 12
h = 12 ÷ 3
h = 4
3. Using Bar Models to Solve Equations
Bar models help us visualise the equation and find the solution.
The key strategy: Let the letter be 1 unit, then find what 1 unit equals!
Example: Buttons in a Container
A container had k yellow buttons at first. After Mrs Tan put the same number of yellow buttons (k) and 4 green buttons into the container, there were 20 buttons in all. How many buttons were there in the container at first?
Setting up the equation:
k + k + 4 = 20, which is 2k + 4 = 20
Solving step by step:
2k + 4 = 20
2k = 20 - 4 (subtract 4 from both sides)
2k = 16
k = 16 ÷ 2 (divide both sides by 2)
k = 8
There were 8 buttons in the container at first.
Keeping Both Sides Balanced
Think of the balanced scale: whatever we do to one side of 2k + 4 = 20, we must do to the other side too.
After subtracting 4 from both sides:
2k = 16
After dividing both sides by 2:
k = 8
Practice: Bar Model Equation
Lucy had 18 coins in her savings jar at first. When she put p coins in the jar, there were 35 coins in all. How many coins did she put in the jar?
ShowHide solution
18 + p = 35
p = 35 - 18
p = 17
Lucy put 17 coins in the jar.
4. Multi-Step Equations
Some equations need two or more steps to solve.
Strategy: First get the letter term by itself (for example, 2x = 56), then find 1 unit (x = 28).
Example: Table Tennis and Badminton
There is a total of 70 children playing table tennis or badminton in a sports hall. There are x children playing table tennis. There are 14 more children playing badminton than table tennis. How many children are playing table tennis?
Setting up the equation:
Badminton = x + 14
x + x + 14 = 70
2x + 14 = 70
Solving step by step:
2x + 14 = 70
2x = 70 - 14
2x = 56
x = 56 ÷ 2
x = 28
28 children are playing table tennis.
(Check: 28 + (28 + 14) = 28 + 42 = 70 ✓)
Example: Sum of Two Numbers
The sum of two numbers is 60. The smaller number is p. The greater number is 4 times the smaller number. What are the two numbers?
Setting up:
p + 4p = 60
5p = 60
Solving:
5p = 60
p = 60 ÷ 5
p = 12
Greater number = 4p = 4 × 12 = 48
The smaller number is 12. The greater number is 48.
(Check: 12 + 48 = 60 ✓, and 48 = 4 × 12 ✓)
Practice: Multi-Step Equation
Khadijah baked n chocolate cookies, 30 coconut cookies, and 2n peanut cookies. She baked 75 cookies altogether. How many chocolate cookies did she bake?
ShowHide solution
n + 30 + 2n = 75
3n + 30 = 75
3n = 75 - 30
3n = 45
n = 45 ÷ 3
n = 15
She baked 15 chocolate cookies.
5. Word Problems with Equations
Example: James and Alex's Money
James has $x. He has $14 less than Alex. They have $50 in all. How much money does James have?
Understanding the problem:
James has $x
"James has $14 less than Alex" means Alex has $14 more
So Alex has $(x + 14)
Solving:
x + x + 14 = 50
2x + 14 = 50
2x = 50 - 14
2x = 36
x = 36 ÷ 2
x = 18
James has $18.
(Check: James = $18, Alex = $18 + $14 = $32, Total = $18 + $32 = $50 ✓)
Practice: Word Problem
The sum of two numbers is 96. The second number is r. The first number is 7 times the second number. What is the first number?
ShowHide solution
Second number = r
First number = 7 × r = 7r
7r + r = 96
8r = 96
r = 96 ÷ 8
r = 12
First number = 7r = 7 × 12 = 84
(Check: 84 + 12 = 96 ✓, and 84 = 7 × 12 ✓)
Challenge: Complex Word Problem
Andy's mass is (j + 3) kg. Bill is j kg heavier than Andy. The total mass of the two boys is 45 kg. Find the value of j.
ShowHide solution
Andy = (j + 3) kg
Bill = Andy + j = (j + 3) + j = (2j + 3) kg
(j + 3) + (2j + 3) = 45
3j + 6 = 45
3j = 45 - 6
3j = 39
j = 39 ÷ 3
j = 13
(Check: Andy = 13 + 3 = 16 kg, Bill = 16 + 13 = 29 kg, Total = 16 + 29 = 45 kg ✓)
Steps to Solve Equations
- Read the problem carefully and identify the unknown
- Draw a bar model if it helps you see the problem
- Write the equation
- Simplify by adding the letter terms together (for example, x + x = 2x)
- Get the letter term by itself on one side of the equals sign
- Solve to find the value of the letter
- Check your answer by substituting back!
Key Takeaways
- ✓ An equation has an equals sign (=)
- ✓ Solving means finding the value of the unknown
- ✓ Bar models help visualise equations
- ✓ In an equation like 9g = 72, finding g is like finding what 1 unit equals
- ✓ Whatever you do to one side, do to the other side
- ✓ Always check your answer by substituting back!