in order to find out where the equations intersect, then we have to see when they equal each other. This is done by subistituting one of they y values into the other.
So we will see what y equals by isolating it in both equations.
y - 4x = 12
y = 4x + 12
and other equation
2 - y = 2(x + 2)^2
-2(x+2)^2 + 2 = y
so now that we have both the y's isolated we consider, as stated earlier, that in order to intersect, the equations have equal x values and equal y values. So we know that y = y and x = x. Since y = y, we can say that both equations contain sides opposite y that are equal. So:
4x + 12 = -2(x+2)^2 + 2
Now we solve for x.
4x + 12 = -2(x^2 + 4x + 4) + 2
4x + 12 = -2x^2 - 8x - 8 + 2
2x^2 + 12x + 18 = 0
2(x^2 + 6x + 9) = 0
2(x + 3)^2 = 0
x = -3
now we know that the lines intersect at only the point x = -3. We use this value in an equation to get y. So
y = 4(-3) + 12
y = -12 + 12
y = 0
The answer is (-3, 0). however we can use this value in the other equation to check to see if it is correct.
0 = -2(-3 + 2)^2 + 2
0 = -2(-1)^2 + 2
0 = -2 + 2
0 = 0
since 0 = 0 is a true statement then we have the right answer.
