Answer:
The two intersections are (3/2 , 5/4) and (4,0).
I put two ways to do it. You can pick your favorite of these are try another route if you like.
Step-by-step explanation:
The system is:
![y=x^2-6x+8](https://tex.z-dn.net/?f=y%3Dx%5E2-6x%2B8)
.
I don't know how good at factoring you are but the top equation consists of polynomial expression that has a factor of (x-4). I see that if I solve 2y+x=4 for 2y I get 2y=-x+4 which is the opposite of (x-4) so -2y=x-4.
So anyways, factoring x^2-6x+8=(x-4)(x-2) because -4+(-2)=-6 while -4(-2)=8.
This is the system I'm looking at right now:
![y=(x-4)(x-2)](https://tex.z-dn.net/?f=y%3D%28x-4%29%28x-2%29)
![-2y=x-4](https://tex.z-dn.net/?f=-2y%3Dx-4)
I'm going to put -2y in for (x-4) in the first equation:
![y=-2y(x-2)](https://tex.z-dn.net/?f=y%3D-2y%28x-2%29)
So one solution will occur when y is 0.
Now assume y is not 0 and divide both sides by y:
![1=-2(x-2)](https://tex.z-dn.net/?f=1%3D-2%28x-2%29)
Distribute:
![1=-2x+4](https://tex.z-dn.net/?f=1%3D-2x%2B4)
Subtract 4 on both sides:
![-3=-2x](https://tex.z-dn.net/?f=-3%3D-2x)
Divide both sides by -2:
![\frac{-3}{-2}=x](https://tex.z-dn.net/?f=%5Cfrac%7B-3%7D%7B-2%7D%3Dx)
![\frac{3}{2}=x](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B2%7D%3Dx)
![x=\frac{3}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B3%7D%7B2%7D)
Now let's go back to one of the original equations:
2y=-x+4
Divide both sides by 2:
![y=\frac{-x+4}{2}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B-x%2B4%7D%7B2%7D)
Plug in 3/2 for x:
![y=\frac{\frac{-3}{2}+4}{2}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B%5Cfrac%7B-3%7D%7B2%7D%2B4%7D%7B2%7D)
Multiply top and bottom by 2:
![y=\frac{-3+8}{4}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B-3%2B8%7D%7B4%7D)
![y=\frac{5}{4}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B5%7D%7B4%7D)
So one solution is at (3/2 , 5/4).
The other solution happened at y=0:
2y=-x+4
Plug in 0 for y:
2(0)=-x+4
0=-x+4
Add x on both sides:
x=4
So the other point of intersection is (4,0).
-------------------------------------------------------
The two intersections are (3/2 , 5/4) and (4,0).
Now if you don't like that way:
![y=x^2-6x+8](https://tex.z-dn.net/?f=y%3Dx%5E2-6x%2B8)
![2y+x=4](https://tex.z-dn.net/?f=2y%2Bx%3D4)
Replace y in bottom equation with (x^2-6x+8):
![2(x^2-6x+8)+x=4](https://tex.z-dn.net/?f=2%28x%5E2-6x%2B8%29%2Bx%3D4)
Distribute:
![2x^2-12x+16+x=4](https://tex.z-dn.net/?f=2x%5E2-12x%2B16%2Bx%3D4)
Subtract 4 on both sides:
![2x^2-12x+16+x-4=0](https://tex.z-dn.net/?f=2x%5E2-12x%2B16%2Bx-4%3D0)
Combine like terms:
![2x^2-11x+12=0](https://tex.z-dn.net/?f=2x%5E2-11x%2B12%3D0)
Compare this to ![ax^2+bx+c=0](https://tex.z-dn.net/?f=ax%5E2%2Bbx%2Bc%3D0)
![a=2](https://tex.z-dn.net/?f=a%3D2)
![b=-11](https://tex.z-dn.net/?f=b%3D-11)
![c=12](https://tex.z-dn.net/?f=c%3D12)
The quadratic formula is
![x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-b%20%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
{Plug in our numbers:
![x=\frac{11 \pm \sqrt{(-11)^2-4(2)(12)}}{2(2)}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B11%20%5Cpm%20%5Csqrt%7B%28-11%29%5E2-4%282%29%2812%29%7D%7D%7B2%282%29%7D)
![x=\frac{11 \pm \sqrt{121-96}}{4}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B11%20%5Cpm%20%5Csqrt%7B121-96%7D%7D%7B4%7D)
![x=\frac{11 \pm \sqrt{25}}{4}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B11%20%5Cpm%20%5Csqrt%7B25%7D%7D%7B4%7D)
![x=\frac{11 \pm 5}{4}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B11%20%5Cpm%205%7D%7B4%7D)
![x=\frac{11+5}{4} \text{ or } \frac{11-5}{4}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B11%2B5%7D%7B4%7D%20%5Ctext%7B%20or%20%7D%20%5Cfrac%7B11-5%7D%7B4%7D)
![x=\frac{16}{4} \text{ or } \frac{6}{4}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B16%7D%7B4%7D%20%5Ctext%7B%20or%20%7D%20%5Cfrac%7B6%7D%7B4%7D)
![x=4 \text{ or } \frac{3}{2}](https://tex.z-dn.net/?f=x%3D4%20%5Ctext%7B%20or%20%7D%20%5Cfrac%7B3%7D%7B2%7D)
Using 2y+x=4 let's find the correspond y-coordinates.
If x=4:
2y+4=4
Subtract 4 on both sides:
2y=0
Divide both sides by 2:
y=0
So we have (4,0) is a point of intersection.
If x=3/2
2y+(3/2)=4
Subtract (3/2) on both sides:
2y=4-(3/2)
2y=5/2
Divide 2 on both sides:
y=5/4
The other intersection is (3/2 , 5/4).