Coordinate Geometry: Line Intersection

Now, you give it a try:

      Which of the following coordinates describes the intersection point of the linear equations y = -x+4 and y = 2x+7?

Incorrect.

First set the two equations equal to each other:

         -x+4 = 2x+7

-->    -x+4 = 2x+7  [Subtract 2x from both sides.]

-->    -3x+4 = 7  [Subtract 4 from both sides.]

-->    -3x = 3  [Divide both sides by -3.]

-->    x = -1

POE all answer choices that do not have an x value of -1. Then solve for y.

Incorrect.

First set -x+4 equal to 2x+7 and solve for x. Then, substitute that value into one of the original equations to solve for y.

You must have missed a negative sign somewhere...

When lines in a coordinate system intersect, for a brief moment they share the same coordinates. In other words, they have the same x and y at the point of intersection.

Consider the following example:

What is the intersection point of the linear equations y = -2x+10 and y = x-2?

Since the lines share the same coordinates at the point of intersection, equate the linear equations like so:

-2x+10 = x-2;

Then solve:

-->    -2x+10 = x-2 [Subtract x from both sides.]

-->    -3x+10 = -2 [Subtract 10 from both sides.]

-->    -3x = -12 [Divide both sides by -3.]

-->    x = 4

Thus the x-coordinate is 4.

To find the y-coordinate, plug in the x-value using one of the linear equations. In the example above, plug x = 4 into the linear equation y = x-2 to get:

     y = (4)-2

--> y = 2

The point of intersection of y = -2x+10 and y = x-2 is (4,2).

Correct.

First set the two equations equal to each other:

         -x+4 = 2x+7

-->    -x+4 = 2x+7  [Subtract 2x from both sides.]

-->    -3x+4 = 7  [Subtract 4 from both sides.]

-->    -3x = 3  [Divide both sides by -3.]

-->    x = -1

Then substitute the obtained x value into one of the original equations to solve for y:

        y = -(-1) +4

-->   y = 1+4

-->   y = 5

Therefore, the intersection point is (-1,5).