How do you find the coordinate of where two lines intersect?

Question:

Line A has a gradient of 4 and passes through point (5,6).

Line B passes through points C (0,5) and D (2,0).

Find the coordinates of the point where the two lines intersection.

 

Solution:

First of all find the equation of line A:

Using y= mx + c,

Applying the gradient, line A has equation, y = 4x + c

To find c, substitute in the coordinates of point P,

6 = (4x5) + c

6 = 20 + c

c = 6 - 20 = -14

Therefore the equation of line A is y = 4x - 14

 

Now find the equation of line B:

Using ( y₂ - y₁ ) / ( x₂ - x₁ ) = gradient of a line

Substitute in coordinates of points C and D,

( y_C - y_B ) / ( x_C - x_B ) = ( 5 - 0 ) / ( 0 - 2 ) = 5/-2 or -5/2

Using y = mx + c

Applying the gradient found, line B has the equation, y = -5/2 x + c

To find c, substitute in the coordinates of point C,

5 = ( -5/2 x 0 ) + c

c = 5

Therefore the equation of line B is y = -5/2 x + 5

This can be rearranged,

(multiply everything by 2) --> 2y = -5x + 10

(rearrange) ---> 5x + 2y = 10

You can check your answer by using the coordinates of point D,

( 5 x 2 ) + ( 2 x 0 ) = 10 ---> Yes

 

Finally find the coordinates where the lines intersect:

A   y = 4x - 14

B   5x + 2y = 10

A x2  2y = 8x - 28

Rearrange 8x - 2y = 28

Using simultaneous equations, add A x2 and B, to eliminate y,

5x + 8x + 2y - 2y = 10 + 28

13x = 38

x = 38/13

Substitute in x to A to find y,

y = ( 4 x 38/13 ) - 14

y = 152/13 - 182/13

y = -30/13

Put these coordinates into the equation for line B to check it works,

( 5 x 38/13 ) + (2 x -30/13 ) = 10

190/13 - 60/13 = 130/13 = 10 ----> Yes

 

Answer:

The lines cross at coordinate ( 38/13, -30/13 )