Straight Lines

Straight Line Graphs

A general linear function has the form y = mx + c , where m, c are constants.

Example 1

A straight line passes through the two points P ( x, y ) and Q ( x, y ) with coordinates P (0 , 2) and Q (1 , 5) . Find the equation of this straight line.

Solution

The general equation of a straight line is y = mx + c. Since the line passes through the points P , with coordinates x = 0 , y = 2, and Q , with coordinates x = 1, y = 5, these coordinates must satisfy this equation, i.e.

2 = m × 0 + c

1 = m × 5 + c

Solving these equations gives c = 2 and m = 3 , i.e. the line is y = 3 x + 2 .

Exercise 1

In each of the following find the equation of the straight line through the given pairs of points.

(a) The points P (0 , - 3) and Q (2 , 1) .

(b) The points P (0 , 4) and Q (1 , 3) .

Solution

(a) The general equation of a straight line is y = mx + c . Since the line passes through the points P and Q , the coordinates of both points must satisfy this equation. The point P has coordinates x = 0 , y = - 3 and the point Q has coordinates x = 2 , y = 1 . These satisfy the pair of simultaneous equations

- 3 = m × 0 + c

1 = m × 2 + c

Solving these equations gives c = - 3 and m = 2 , i.e. the line is y = 2 x - 3 .

(b) The general equation of a straight line is y = mx + c . Since the line passes through the points P and Q , the coordinates of both points must satisfy this equation. The point P has coordinates x = 0 , y = 4 and the point Q has coordinates x = 1 , y = 3. These satisfy the pair of simultaneous equations

4 = m × 0 + c

3 = m × 1 + c

Solving these equations gives c = 4 and m = - 3 , i.e. the line is y = - 3 x + 4 .

Gradient of a Straight Line

The gradient of a straight line is defined as follows. Suppose that two points P, Q, on the line have coordinates P( x₁ , y₁ ) and Q( x₂ , y₂ ) .

The gradient of the line is (see diagram above)

Exercise 2

Find the gradient of the line through the points P(3, 9) and Q(2, 3).

Solution

For P(3 , 9) , Q(2 , 3) , the gradient is given by

so that m = 6 in this case.

Intercepts of a Straight Line

By putting x = 0 into the equation y = mx + c , the point where the straight line crosses the y axis is found to be y = c . This is known as the intercept on the y axis . The intercept on the x axis, i.e. when y = 0, is at

0 = mx + c

- c = mx

- c/m = x .

The x and y intercepts.

Example 2

By rearranging the equation 2 y - 3 x - 5 = 0 , show that it is a straight line and find its gradient and intercept. Sketch the line.

Solution

Rearranging the equation,

3 y - 2 x - 5 = 0

3 y = 2 x + 5

(Equation of a straight line with m = 2 / 3 and c = 5 / 3.)

Exercise 3

The equation 2 y - 2 x + 3 = 0 represents a straight line. By rearranging it, find its gradient and the intercepts on the y and x axis. Sketch the straight line it represents.

Solution

For the equation 2 y - 2 x + 3 = 0,

2 y - 2 x + 3 = 0

2 y = 2 x - 3

so that m = 1 and c = - 3 / 2. The intercept on the x axis is - c/m = - ( - 3 / 2) / 1 = 3 / 2.

Quiz

A straight line has the equation 3 x + y +3 = 0. If P is the point where the line crosses the x-axis and Q is the point where it crosses the y-axis, which of the following pair is P,Q?

Solution

The line crosses the x axis when y = 0 . Putting this into the equation of the line, 3 x + y + 3 = 0 , gives

3 x + 0 + 3 = 0

3 x = - 3 x = - 1.

Thus P ( - 1 , 0) is the first point.

The line crosses the y axis when x = 0. Putting this into the equation of the line,

3 × (0) + y + 3 = 0

y + 3 = 0 y = - 3.

Thus Q (0 , - 3) is the second point.