Specific Learning Outcomes
You are required to:
- Form linear equations in one unknown in different situations
- Solve linear equations in one unknown in different situations
- Apply linear equations in one unknown to real-life situations
- Reflect on the use of linear equations in real-life situations
What Is Covered
- Meaning of linear equations
- Formation of linear equations
- Solving linear equations
- Applications of linear equations
2.2 Linear Equations
Linear equations are equations in which the variable is raised to power one.
They are also called first order equations.
Linear equations of one variable take the form:
ax + b = 0, where a and b are numbers and x is a variable.
Examples:
- 2x + 3 = 0
- 4x = 8
Note:
Equations have an equal sign (=) while expressions do not.
2.2.1 Formation of Linear Equations
Example 2.6
The length of a rectangle is 4 cm more than its width.
If the width is x cm and the perimeter is 24 cm,
form an equation.
Solution
Width = x cm
Length = x + 4 cm
Perimeter = 2(length + width)
2(x + 4 + x) = 24
2(2x + 4) = 24
2.2.2 Solution to Linear Equations
Example 2.7
Solve: 4x − 9 = 19
Solution
4x = 19 + 9
4x = 28
x = 7
Example 2.8
Solve: ½x + 5 = 8
Solution
Multiply both sides by 2
x + 10 = 16
x = 6
2.2.3 Applications of Linear Equations
Example 2.9
In one week, Kimari spends x minutes on the internet.
Kyalo spends twice the amount of time spent by Kimari while
Wafula spends 10 minutes more than Kyalo.
If the total time they spend was 120 minutes,
how many minutes were spent by Kimari on the internet?
Solution
Time used by Kimari = x
Time used by Kyalo = 2x
Time used by Wafula = 2x + 10
x + 2x + 2x + 10 = 120
5x + 10 = 120
5x = 110
x = 22
⚖️ Activity: Balance the Equation
4x + 8 = 24
Exercise 2.2
1. Solve the following linear equations:
(a) 5x − 2 = 18
5x − 2 = 18
5x = 20
x = 4
(b) 3x + 7 = 25
3x + 7 = 25
3x = 18
x = 6
(c) 2x − 5 = 9
2x − 5 = 9
2x = 14
x = 7
(d) x ÷ 4 = 6
x ÷ 4 = 6
x = 6 × 4
x = 24
(e) ½x + 3 = 11
½x + 3 = 11
½x = 8
x = 16
2. The length of a rectangle is twice its width.
If the perimeter is 15 cm, find the width.
Let width = x
Length = 2x
Perimeter = 2(x + 2x)
6x = 15
x = 2.5 cm
3. The figure below shows the dimensions of a triangle.
If the perimeter is 20 cm, find x.
(x + 5) + (x + 2) + (2x − 3) = 20
4x + 4 = 20
4x = 16
x = 4
4. Calculate the area of the rectangle shown below.
Length = x + 5
Width = 2x − 3= x + 2
Use the equation on the width to find x
x = 5
Substitute x = 5
Area = 10 × 7 = 70 cm²
5. Fred is 21 years old. Ann is x years younger than Fred.
If the sum of their ages is 35, find x.
Fred = 21
Ann = 21 − x
21 + (21 − x) = 35
x = 7