Topic 11: Equations of Lines

By the end of this topic, you will be able to:

form linear equations with given points draw the graph of a line given its equation

Keywords

1. general form
2. intercept
3. linear equation
4. parallel line
5. perpendicular line
6. slope
7. Equations

Introduction

Suppose you are planning for a ceremony and you are not sure of the total number of guests. How can you easily estimate the number of guests you can accommodate using the resources available? What of transport fees in relation to distance traveled?

In this section you will understand and use linear equations and their graphs.

Forming Linear Equations with given Points

You studied about the Cartesian Plane in Topic 4; and about forming the general formula relating the number of matchsticks and the figure number in Topic 6. The following activity will help you remember the above concepts and also link you to this topic.

Activity 11.1 Forming linear equations (work in groups)

What you need: a graph paper, ruler, pencils, Pens.

What to do:

1. Draw Cartesian axes on the graph paper.

2. Draw any line on a Cartesian plane on your graph paper.

3. Let each member of your group choose a point on the line and record its coordinates.

4. Copy and complete the table below showing the coordinates of the points chosen.

5. Choose any two co-ordinate points.

6. Find the difference between the two x-coordinates and also the difference between the y-coordinates.

7. Multiply the x-values by the difference between y-values and y-values by difference in x-values and obtain a difference.

8. Determine the equation of the line.

9. Present your work to the whole class

Example.

Or 3x-2y=-4

3x-2y+4=0

Therefore, the equation of the line (linear equation) for the line passing through A and B is 2y – 3x-4=0

Or 3x-2y + 4 = 0

Exercise 11.1

1. Form equations passing through the following points.

a) (3, 1), (4, 1), (3, 1)

b) (-7, 2), (4, 0) and (1, 2)

c) (6, 7), (2, 4) and (2, -1)

2. Draw a Cartesian plane and create a triangle using three straight lines. Give their equations.

3. Find the equation of at least 5 straight lines that pass through the point (1, 1).

4. Imagine you are a Town Council Roads Engineer and you intend to open a straight road linking the town council offices to other parts of the town. For the information collected a trader at (-1, -3), a home at (1, 3) and a school at (3, 9) from the town council offices are to be affected. Form the equation of a line that can be used to ascertain the other units which are affected?

Drawing the Graph of a Line given its Equation

You have learnt how to form a linear equation of a line given its points in the previous section. The equation of a straight line is in the form y = mx + c where m is the slope (or gradient) and c is the y-intercept. In this section, you will now learn how to draw a graph given a linear equation.

Activity 11.2 Drawing a graph of a line given its equation (work in groups)

What you need: graph paper, pen, pencil, ruler.

What to do:

1. Consider equation y = x – 2 which is in form of y = mx + c.

2. Use the equation to find the y-coordinates for x = -3, -2, -1, 0, 1 and 2. Use your results to complete the table below.

3. Plot the coordinates above on a Cartesian plane and join them to form a line.

4. Repeat the steps above for different values of m and c.

5. Present your work to the class.

Exercise 11.2

1. Draw the graph of the following lines.

a) y = x + 3 b) 2y = 4x-4 c) y 2x-3= 0

2. Draw the following lines on the same Cartesian plane.

a) i) y = 2x + 6 ii) y = 2x + 3 iii) 2y = x + 4

b) Identify the two lines which are parallel and find their gradient

c) What do you observe?