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Point Gradient Formula

The point gradient formula involves finding the equation of a straight line given one of the points on the line, and the gradient of its line.

We know that the gradient of any line is given by m = {y_2~-~y_1}/{x_2~-~x_1}

Hence y2 – y1 = m(x2 – x1)point_gradient_formula

Now given a point (x1 – y1) and slope m, we get

m = {y~-~y_1}/{x~-~x_1} or

y – y1 = m(x – x1)

This equation is called the point gradient formula for the equation of a straight line passing through (x1, y1) with gradient m.

 

Example 1: Find the equation of a straight line that passes through (1, 0) and has a gradient 2.

(x1, y1) is (1, 0) and m = 2

So the equation of the line is –

(y – y1) = m(x – x1)

y – 0  =  2(x – 1)

y = 2x -2

 

Example 2: Find the value of c if the given point (2, 3) lies on the line y = 2x + c.

Substitute the point (2, 3) in the equation y = 2x + c (x = 2, and y = 3).

3 = 2 x 2 + c

3 = 4 + c

So c = -1

The equation of the line is y = 2x – 1

 

Example 3: Find the equation of a line that has a gradient of 2 and passes through the midpoint of (2, 5) and (-4, 3).

First we must find the midpoint of (2, 5) and (-4, 3).

The midpoint formula is ({x_1~+~x_2}/2,~{y_1~+~y_2}/2)

We get the midpoint as ({2~-~4}/2,~{5~+~3}/2)

=  (-2/2,~8/2)

=  (-1, 4)

So the line passes through (-1, 4) and has a slope or gradient of 2.

Using point gradient formula of a line, we get –

(y – y1) = m(x – x1)

(y – 4)  =  2(x – (-1)), since (x1 , y1) = (-1, 4) and m = 2

y – 4 = 2(x + 1)

y – 4 = 2x + 2

y = 2x + 6

Or writing the above equation in general form, it will be 2x – y + 6 = 0