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Year 10 Interactive Maths - Second Edition


Equation of a Straight Line

To establish a rule for the equation of a straight line, consider the previous example.

An increase in distance by 1 km results in an increase in cost of $3. We say that the rate of change of cost with respect to distance is $3 per kilometre.

The information given in the graph can be represented by the equation c = 5 + 3d.  That is:

c = 3d + 5 where the gradient of the line = coefficient of d = 3 and the vertical axis intercept = 5


In general:

A line with equation y = mx + c has gradient m and y-intercept c.

y = mx + c where m is the gradient and c is the y-intercept

The gradient of a straight line is the coefficient of x.


Particular Case

If a straight line passes through the origin, then its y-intercept is 0.  So, the equation of a straight line passing through the origin is

y = mx

where m is the gradient of the line.

The graph of y = mx which passes through the origin at (0,0).


Example 7

Write down the gradient and the y-intercept for the following equations:  (a)  y = 4x + 3    (b)  6x + 3y = 9

Solution:

(a)  Comparing y = 4x + 3 with y = mx + c gives m = 4, c = 3.  So, the gradient is 4 and the y-intercept is 3.              

(b)  Write 6x + 3y = 9 in the form y = mx + c

Subtract 6x from both sides and then divide both sides by 3 to find y = -2x + 3.

Comparing y = -2x + 3 with y = mx + c gives m = -2, c = 3. So, the gradient is -2 and the y-intercept is 3.


Example 8

Write down the equation of the straight line that has m = 5 and c = 3.

Solution:

The general equation of the straight line is y = mx + c. Substituting m = 5 and c = 3 gives y = 5x + 3.                 


Example 9

Calculate the gradient of the straight line given in the following diagram; and find its equation.

The graph of the straight line joined by the points (0,5) and (3,11).

Solution:

Let (x1, y1) = (0, 5) and (x2, y2) = (3, 11). m = (y2 - y1) / (x2 - x1) = (11 - 5) / (3 - 0) = 6 / 3 = 2 and c = 5. The general equation of the straight line is y = mx + c. Substituting m = 2 and c = 5 gives y = 2x + 5.


Example 10

Find the equation of the line joining the points (2, 3) and (4,7).

Solution:

The graph of the straight line joined by the points (2,3) and (4,7).

Let (x1, y1) = (2,3) and (x2, y2) = (4, 7). m = (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4 / 2 = 2. The general equation of a straight line is y = mx + c. Therefore, y = 2x + c. Using point (2,3) which is on the line in y = 2x + c gives c = -1. So, the equation is y = 2x - 1.


Key Terms

equation of a straight line, gradient, y-intercept


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