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Slope Intercept Form of a Straight Line

Any straight line (or linear equation) can be expressed as – slope_intercept_form_of_a_straight_line

y = mx + c, where

m is the slope (or gradient) of the line, and c is the y-intercept of the line i.e. where point where the line meets the y-axis.

 

We also saw earlier that the slope of a line can be given by m = tan theta, where theta is the angle of inclination of the line with the x-axis in positive direction.

 

Here are a few characteristics of the slope intercept form of a straight line:

  • To find the x and y intercepts of a linear equation representing a straight line, we put x = 0 to get the y-intercept, and put y = 0 to get the x-intercept. These two points are where the line meets the x and y axis respectively.
  • If c = 0, then y = mx. This means the line passes through the origin.
  • If m = 0, then y = c. This means the line is parallel to the x-axis (the horizontal axis).
  • If the slopes (gradients) or two lines are the same, then the two lines are parallel. In other words, parallel lines have the same gradient.

 

The general equation of a line is given by  ax  +  by  +  c  =  0, where a, b, and c are integers.

This equation can be rearranged to get  ax  +  c  =  -by

Dividing the equation by -b will give –  {a/-b}x  +  {c/-b}  =  y

y  =  – {a/b}x  +  – {c/b}

This is of the form y  =  mx  +  c, where m  =  – {a/b} and c  =  – {c/b}

 

Now let us look at a few examples. Find the gradient and y-intercept for the lines using the slope intercept form of a straight line.

Example 1:  y = 2x + 3

Comparing this line with y = mx + c, we get m = 2 and c = 3.

So the gradient m is 2 and the y-intercept = c = 3.

So the line meets the y-axis at 3.

 

Example 2:  y = -2x + 3

Again, comparing this line with y = mx + c, we get m = -2 and c = 3.

So the gradient m is 2 and the y-intercept = c = 3.

So the line meets the y-axis at 3, but has a negative gradient i.e. slopes downwards compared to the line in example 1.

 

Example 3:  y = -3x – 3

Again, comparing this line with y = mx + c, we get m = -3 and c = -3.

So the gradient m is -3 and the y-intercept = c = -3.

 

Example 4:  2x + y + 5 = 0

First, we need to rearrange this equation to the equation y = mx + c.

2x + y + 5 = 0

Subtract y from both sides of the equation – 2x + y + 5 – y = 0 – y

We get 2x + 5 = -y.

Now dividing this equation by -1 on both sides, {2x~+~5}/-1~=~{-y}/{-1}, we get

y = -2x – 5

Now compare this with the equation y = mx + c, get

gradient = m = -2

y-intercept = c = -5

 

Example 5:  2x – 3y + 6 = 0

Once again, we need to rearrange this equation to the form y = mx + c.

2x – 3y + 6 = 0

Adding 3y from both sides of the equation – 2x – 3y + 6 + 3y = 0 + 3y

We get 2x + 6 = 3y.

Now dividing this equation by 3 on both sides, {2x~+~6}/3~=~{3y}/3, we get

y = 2/3x + 2

Now compare this with the equation y = mx + c, get

gradient = m = 2/3

y-intercept = c = 2

 

Example 6: The gradient of a line is 3 and its y-intercept is -2. Write the equation of the straight line in slope intercept form.

Given m = 3 and c = -2, the equation of the straight line is y = mx + c

The equation of the above line is y = 3x – 2

 

Here are a few more examples of slope intercept form of a straight line.