The Straight Line on Coordinate Geometry SS1 Further Mathematics Lesson Note

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  =  MB Increase in x from A to B =  AM

Example 

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

The gradient of PQ = 

Increase in y   =    –AQ 

Increase in x             PA

= -4    =    -1

    8            2  

Example 2

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

Re-arrange the equation: 

4y = – 7x + 8

y =  + 2

Therefore, gradient (m) =  

y – 7/4 intercept (c) = 2 

SKETCHING GRAPHS OF STRAIGHT LINES

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

2x + 3y = 6, gradient =  , y-intercept(c) = 2

Example

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

Solution

When x = 0 ,- 3y = 12

  y = – 4

The line crosses the y–axis at (0, – 4).

When y = 0 , 4x = 12

x = 3

The line crosses the x-axis at (3, 0).

From the graph:

Gradient m = y2 – y1

                       x2 – x1   i.e. 

0 + 4

3 + 0                                                                                   y-intercept = – 4

Lines Parallel To Axis

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

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

Notice that the equation of the y–axis is x = 0

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 and passes through the point (- 3, 2).

Solution

Using the formula y – y1 = m(x – x1)

Where (x1, y1) = (- 3, 2) and m =   

y – 2 =  (x + 3) 

3y – 6 = – x – 3 

x + 3y = 3

EQUATION OF A LINE

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

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

y = 3x – 4 

y = – ½x – 3 

Solution:

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

Gradient is – ½ , intercept on y-axis

GRADIENT AND ONE-POINT FORM

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

y – y1 = m(x -x1)

m = 2; x1 = 3; 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  

ANGLE OF SLOPE

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 = 8 

       7 – 3      4

= 2

Let the angle of slope of the line; tan o = 2

= 63.43°