10-Week Mathematics Catch-Up Programme (CAPS)

Week 3 – Day 4

Finding the Equation of a Straight Line

Duration: 1 Hour

Grade Level: Grade 9 Foundation (Preparing for Grade 10 & 11 Functions)

Learning Outcomes

By the end of today's lesson you should be able to:

• Recognise the form (y = mx + c).
• Identify the gradient and y-intercept from an equation.
• Write the equation of a line from its gradient and y-intercept.
• Write the equation of a line from a graph.
• Write the equation of a line from a point and a gradient.

Part A – Study Guide

1. The General Equation of a Straight Line

Every straight line can be written as:

$\boxed{y = mx + c}$

Where:

• (m) = gradient (slope)
• (c) = y-intercept (where the line crosses the y-axis)

Example 1

$y = 2x + 3$

Gradient = 2

y-intercept = 3

Example 2

$y = -4x + 1$

Gradient = -4

y-intercept = 1

Example 3

$y = x - 5$

Remember:

$y = 1x - 5$

Gradient = 1

y-intercept = -5

2. Writing an Equation from a Gradient and y-Intercept

Example:

Gradient = 3

y-intercept = -2

Substitute into the form $y = mx + c$:

$y = 3x - 2$

Another example:

Gradient = -1

y-intercept = 4

$y = -x + 4$

3. Finding the Equation from a Graph

Suppose a graph:

• Crosses the y-axis at (2).
• Rises by (1) for every (1) moved to the right.

Gradient = 1

y-intercept = 2

Equation:

$y = x + 2$

4. Finding the Equation from a Point and a Gradient

Suppose:

• Gradient = (2)
• The line passes through ((0,5))

Because ((0,5)) is on the y-axis, the y-intercept is 5.

Equation:

$y = 2x + 5$

Another Example

Gradient = (-3)

Point = ((0,-2))

Equation:

$y = -3x - 2$

5. Checking Your Equation

Always substitute a known point into your equation.

Example:

Equation:

$y = 2x + 1$

Does the point ((2,5)) lie on the line?

Substitute:

$y = 2(2) + 1 = 5$

Yes, the point lies on the line.

Common Mistakes

Mistake 1

Forgetting the sign of the y-intercept.

Example:

$y = 3x - 4$

The y-intercept is -4, not 4.

Mistake 2

Writing:

$y = 2 + x$

instead of the standard form:

$y = x + 2$

Both are mathematically correct, but exams usually expect the form (y = mx + c).

Mistake 3

Confusing the gradient with the y-intercept.

The gradient is the number multiplying x.

The y-intercept is the constant term.

Part B – Worked Examples

Example 1

Write the equation of a line with:

• Gradient = 4
• y-intercept = 3

Answer:

$y = 4x + 3$

Example 2

Write the equation of a line with:

• Gradient = -2
• y-intercept = 5

Answer:

$y = -2x + 5$

Example 3

Find the gradient and y-intercept of:

$y = 6x - 7$

Gradient = 6

y-intercept = -7

Example 4

Does the point ((3,8)) lie on the line

$y = 2x + 2?$

Substitute:

$2(3) + 2 = 8$

Yes.

Example 5

Does the point ((2,7)) lie on

$y = 3x + 2?$

Substitute:

$3(2) + 2 = 8$

The y-coordinate should be 8, not 7.

No, the point does not lie on the line.

Part C – Practice Questions

Section A – State the Gradient and y-Intercept

1. $y = 5x + 2$

2. $y = -3x + 7$

3. $y = x - 4$

4. $y = -x - 6$

5. $y = 8x$

6. $y = 2x - 10$

7. $y = -5x$

8. $y = 7x + 9$

9. $y = -4x - 3$

10. $y = 6 - 2x$

Section B – Write the Equation

Write the equation of a line with:

11. Gradient = 3, y-intercept = 4

12. Gradient = -2, y-intercept = -5

13. Gradient = 1, y-intercept = 8

14. Gradient = -6, y-intercept = 2

15. Gradient = 0, y-intercept = 7

Section C – Does the Point Lie on the Line?

Answer Yes or No and show your working.

16. Does ((2,7)) lie on (y = 3x + 1)?

17. Does ((4,9)) lie on (y = 2x + 1)?

18. Does ((-1,4)) lie on (y = -x + 3)?

19. Does ((5,11)) lie on (y = 2x + 1)?

20. Does ((0,-2)) lie on (y = 4x - 2)?

Section D – From Tables to Equations

Look at each table and write the equation.

21.

22.

23.

24.

25.

Challenge Questions

26. Write the equation of a line that:

• has a gradient of 5;
• passes through the point ((0,-3)).

27. Write the equation of a line that:

• has a gradient of -4;
• passes through the point ((0,8)).

28. Which of these lines is steeper?

$y = 7x + 1$

or

$y = -5x + 2$

Explain your answer.

29. A gym charges a joining fee of R150 plus R250 per month.

Let:

• (x) = number of months
• (y) = total amount paid

Write the equation.

What will a member pay after 8 months?

30. A delivery company charges a fixed fee of R50 plus R12 per kilometre.

a) Write the equation.

b) What is the cost for 15 km?

Answers

Section A

Section B

$y = 3x + 4$

$y = -2x - 5$

$y = x + 8$

$y = -6x + 2$

$y = 7$

Section C

$3(2)+1=7$

Yes

$2(4)+1=9$

Yes

$-(-1)+3=4$

Yes

$2(5)+1=11$

Yes

$4(0)-2=-2$

Yes

Section D

Gradient = 2, intercept = 2

$y = 2x + 2$

Gradient = -1, intercept = 5

$y = -x + 5$

Gradient = 2, intercept = -1

$y = 2x - 1$

Constant value:

$y = 6$

Gradient = -3, intercept = -2

$y = -3x - 2$

Challenge Answers

$y = 5x - 3$

$y = -4x + 8$

28. $(y = 7x + 1)$ is steeper because the absolute value of its gradient is larger:

• $|7| = 7$
• $|-5| = 5$

Equation:

$y = 250x + 150$

After 8 months:

$y = 250(8) + 150 = 2,150$

Total = R2 150

a)

$y = 12x + 50$

b)

$y = 12(15) + 50 = 230$

Cost = R230

Parent's Notes

By the end of today's lesson, your student should be able to:

• Recognise the meaning of (m) (gradient) and (c) (y-intercept).
• Write equations in the form (y = mx + c).
• Check whether a point lies on a line by substitution.
• Identify equations from simple tables of values.
• Apply straight-line equations to practical contexts such as taxi fares and membership costs.