The Straight Line on Coordinate Geometry – SS1 Further Mathematics Lesson Note (Second Term)

A straight line on a coordinate plane has a gradient, which is the rate of change of y compared with x, and a y-intercept, the value of y where it crosses the y-axis. Its equation is y = mx + c, where m is the gradient and c is the y-intercept.

Behavioural objectives

By the end of the lesson, students should be able to:

1. find the gradient of a line joining two points;
2. find the gradient and y-intercept of a line from its equation;
3. sketch the graph of a straight line;
4. write the equation of a line parallel to either axis;
5. find the equation of a line from its gradient and one point on it;
6. find the angle of slope of a line.

What is the gradient of a straight line?

The gradient of a straight line is the rate of change of y compared with x.

For example, if the gradient is 2, then for any increase in x, y increases two times as much.

The gradient of AB = (increase in y from A to B) / (increase in x from A to B) = MB / AM

Example 1

Find the gradient of the line joining P(7, -2) and Q(-1, 2).

The gradient of PQ = (increase in y) / (increase in x) = -AQ / PA = -4/8 = -1/2

Example 2

Find the gradient of the line 7x + 4y – 8 = 0.

Re-arrange the equation:

4y = -7x + 8

y = -7/4 x + 2

Therefore, gradient (m) = -7/4 and y-intercept (c) = 2.

How do you sketch the graph of a straight line?

Given the equation y = 3x – 2, gradient = 3 and y-intercept (c) = -2.

For 2x + 3y = 6, gradient = -2/3 and y-intercept (c) = 2.

Example

Sketch the graph of the line whose equation is 4x – 3y = 12.

Solution

When x = 0, -3y = 12, so y = -4. The line crosses the y-axis at (0, -4).

When y = 0, 4x = 12, so x = 3. The line crosses the x-axis at (3, 0).

From the graph, gradient m = (y2 – y1) / (x2 – x1) = (0 + 4) / (3 – 0) = 4/3

The y-intercept is -4.

What are lines parallel to the axes?

Any line parallel to the x-axis has a gradient of zero. The equation of such a line is always in the form y = c, where c may be any number.

Draw a graph of y = 5 and y = -3.

The gradient of a line that is parallel to the y-axis is undefined. The equation of such a line is always in the form x = a, where a may be any number.

The figure will show the graph of the lines x = 2 and x = -4.

Notice that the equation of the y-axis is x = 0.

What is the equation of a straight line?

The equation of a straight line is of the form y = mx + c, where m is the gradient and c is the y-intercept.

Example 1

Determine the equation of a straight line whose gradient is -1/3 and which passes through the point (-3, 2).

Solution

Using the formula y – y1 = m(x – x1), where (x1, y1) = (-3, 2) and m = -1/3:

y – 2 = -1/3 (x + 3)

3y – 6 = -x – 3

x + 3y = 3

Finding the gradient and y-intercept from an equation

The equation of a straight line is given by y = mx + c.

Example: Find the gradient and the intercept on the y-axis of the following lines:

1. y = 3x – 4
2. y = -1/2 x – 3

Solution

Compare y = 3x – 4 with y = mx + c. Hence the gradient is 3 and the intercept on the y-axis is -4.

For y = -1/2 x – 3, the gradient is -1/2 and the intercept on the y-axis is -3.

How do you find the equation of a line from its gradient and one point?

Example: Find the equation of a straight line of slope 2 if it passes through the point (3, -2).

Use y – y1 = m(x – x1), where m = 2, x1 = 3 and y1 = -2.

Hence the equation of the straight line is:

y – (-2) = 2(x – 3)

y + 2 = 2x – 6

y = 2x – 6 – 2 = 2x – 8

y = 2x – 8

How do you find the angle of slope of a line?

Example: Find the gradient of the line joining (3, 2) and (7, 10) and the angle of slope of the line.

Solution

Let m be the gradient of the line. Then

m = (10 – 2) / (7 – 3) = 8/4 = 2

Let θ be the angle of slope of the line. Then tan θ = 2, so θ = 63.43°.

Evaluation

1. State the gradient of a line parallel to the x-axis and the form of its equation.
2. Find the gradient and y-intercept of the line 3x + 2y – 6 = 0.
3. Find the equation of the line with gradient 2 that passes through (1, 4).
4. Find the gradient of the line joining (2, 1) and (4, 7), and the angle of slope of the line.

Assignment

1. Sketch the graph of the line 3x – 2y = 6.
2. Find the equation of the line with gradient -2 that passes through (-1, 3).
3. Draw the graphs of y = 2 and x = -3 on the same axes.