use a equation

Question

Line m passes through the points (6,1) and (2,-3).Line n passes through the points (2,3)and (5,-6).Find the point of intersection of these two lines. 

Answer 1

First we need to find the equation of each line.  To find where they intersect, we set the equations equal and solve for x.  

 

slope of line m = -3 - 1 = -4  = 1           slope of line n = -6 -3 = -9 = -3

                          2 - 6     -4                                           5 - 2    3

 

equation of line m (y = mx + b)            equation of line n

 

use (6,1) to solve for b                        use  (2,3) to solve for b

 

1 = 1 * 6 + b                                       3 = -3 * 2 + b

1 = 6 + b                                             3 = -6 +b

-5 = b                                                 9 = b

 

 

y = x - 5                                             y = -3x +  9

 

where they intersect the y is the same so

 

x - 5 = -3x + 9

4x    = 14

x = 14/4  = 3.5

 

Use either equation to get y

y = 3.5 - 5                           y = -3 * 3.5 + 9

y = -1.5                               y = -10.5 + 9 = -1.5

 

point of intersection (3.5, -1.5)

Answer 2

Calculate the equations for the two lines.

 

standard form is y=mx+b, m is the slope.

 

The first two points are (6,1) and (2,-3).  the slope is rise divided by run.  Rise = -3-1=-4, Run =2-6=-4

Rise/Run =-4/(-4)=1 so m=1 and the line equation is y = x+b, plug in (6,1) to get 1=6+m or m=-5.

 

Equation for line one is  y = x-5, you can verify this is correct by plugging in the value from the other data point (2,-3) 

which gives -3 = 2 - 5, or -3 = -3, which is true.

 

The next two points are (2,3) and (5,-6). the slope is rise divided by run. Rise = -6-3=-9, Run =5-2=3
Rise/Run =-9/3=-3 so m=-3 and the line equation is y = -3x+b, plug in (2,3) to get 3=-3*2+m or m=9.

Equation for line two is y = -3x+9, you can verify this is correct by plugging in the value from the other data point (5,-6)
which gives -6 = -3*5 + 9, or -6 = -6 which is true also.

 

Now to find the intersection we simply set them equal which they are by definition when and where they intersect.

 

y = x-5 = -3x + 9,  ==>  4x = 14,  ==> x = 3.5

 

We plug x=3.5 back into either of the equations to get the y-value of the point of intersection.

 

y = x-5 = 3.5 - 5 = -1.5.   The point of intersection is therefore (3.5, -1.5).

 

We can check this using the second equation, y = -3x + 9,  ==>  -1.5 = (-3)(3.5) + 9,  -1.5 = -10.5 + 9 = -1.5,

 

So it checks out.