Finding the point of intersection

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. I have to show my work.

Answer 1

The question wants us to find the point where lines m and n intersect.  In order to do that, we need to find the equations of line m and line n   in slope-intercept form, which is       y=mx+b.

 

Use the formula for slope to find the slope between the corresponding points for each line.

 

Slope = (y₂ - y₁) / (x₂ - x₁)

 

Line m:

 

slope = (-3 - 1) / (2 - 6)

        = -4 / -4

        = 1

 

 

Plug in the slope into the slope-intercept form of the equation to find b using the 1st coordinate.

 

1 = 1(6) + b

 

1 = 6 + b

 

-5 = b

 

 

Line m  is represented by the equation         y = x - 5

 

 

We do the same for line n.

 

slope = (-6 - 3) / (5 - 2)

        = -9 / 3

        = -3

 

 

3 = -3(2) + b

 

3 = -6 + b

 

9 = b

 

 

Line n is represented by the equation          y = -3x + 9

 

 

Now we can find the point of intersection.  The lines will share a single x-coordinate and y-coordinate.  Since both equation are defined by y, we can equate them.

 

 

x - 5 = -3x + 9

 

 

Solve for x from this equation.

 

Add 3x on both sides of the equation.

 

4x - 5 = 9

 

 

Add 5 on both sides of the equation.

 

4x = 14

 

  x = 3.5

 

 

Substitute this value of x into the equation that represents line m to solve for y.

 

y = 3.5 - 5

 

y = -1.5

 

The point of intersection is then (3.5, -1.5).