Answer:
The two solutions in exact form are:
![(\frac{15+\sqrt{335}}{10},\frac{5-3\sqrt{335}}{10})](https://tex.z-dn.net/?f=%28%5Cfrac%7B15%2B%5Csqrt%7B335%7D%7D%7B10%7D%2C%5Cfrac%7B5-3%5Csqrt%7B335%7D%7D%7B10%7D%29)
and
.
If you prefer to look at approximations just put into your calculator:
![(3.3303,-4.9909)](https://tex.z-dn.net/?f=%283.3303%2C-4.9909%29)
and
.
Step-by-step explanation:
I guess you are asked to find the solution the given system.
I'm going to use substitution.
This means I'm going to plug the second equation into the first giving me:
I replaced the 1st y with what the 2nd y equaled.
Before we continue solving this I'm going to expand the
using the following:
.
![(-3x+5)^2=(-3x)^2+2(-3x)(5)+(5)^2](https://tex.z-dn.net/?f=%28-3x%2B5%29%5E2%3D%28-3x%29%5E2%2B2%28-3x%29%285%29%2B%285%29%5E2)
![(-3x+5)^2=9x^2-30x+25](https://tex.z-dn.net/?f=%28-3x%2B5%29%5E2%3D9x%5E2-30x%2B25)
Let's go back to the equation we had:
After expansion of the squared binomial we have:
![9x^2-30x+25+x^2=36](https://tex.z-dn.net/?f=9x%5E2-30x%2B25%2Bx%5E2%3D36)
Combine like terms (doing the
part:
![10x^2-30x+25=36](https://tex.z-dn.net/?f=10x%5E2-30x%2B25%3D36)
Subtract 36 on both sides:
![10x^2-30x+25-36=0](https://tex.z-dn.net/?f=10x%5E2-30x%2B25-36%3D0)
Simplify the 25-36 part:
![10x^2-30x-11=0](https://tex.z-dn.net/?f=10x%5E2-30x-11%3D0)
Compare this to
which is standard form for a quadratic.
We should see the following:
![a=10](https://tex.z-dn.net/?f=a%3D10)
![b=-30](https://tex.z-dn.net/?f=b%3D-30)
![c=-11](https://tex.z-dn.net/?f=c%3D-11)
The formula that solves this equation for the variable
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)
Plugging in our values for
give us:
![x=\frac{30 \pm \sqrt{(-30)^2-4(10)(-11)}}{2(10)}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B30%20%5Cpm%20%5Csqrt%7B%28-30%29%5E2-4%2810%29%28-11%29%7D%7D%7B2%2810%29%7D)
Simplify the bottom; that is 2(10)=20:
![x=\frac{30 \pm \sqrt{(-30)^2-4(10)(-11)}}{20}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B30%20%5Cpm%20%5Csqrt%7B%28-30%29%5E2-4%2810%29%28-11%29%7D%7D%7B20%7D)
Put the inside of square root into the calculator; that is put
in the calculator:
![x=\frac{30 \pm \sqrt{1340}}{20}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B30%20%5Cpm%20%5Csqrt%7B1340%7D%7D%7B20%7D)
Side notes before continuation:
Let's see if 1340 has a perfect square.
I know 1340 is divisible by 10 because it ends in 0.
1340=10(134)
134 is even so it is divisible by 2:
1340=10(2)(67)
1340=2(2)(5)(67)
1340=4(5)(67)
1340=4(335)
4 is a perfect square so we can simplify the square root part further:
.
Let's go back to the solution:
![x=\frac{30 \pm \sqrt{1340}}{20}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B30%20%5Cpm%20%5Csqrt%7B1340%7D%7D%7B20%7D)
![x=\frac{30 \pm 2 \sqrt{335}}{20}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B30%20%5Cpm%202%20%5Csqrt%7B335%7D%7D%7B20%7D)
Now I see all three terms contain a common factor of 2 so I'm going to divide top and bottom by 2:
![x=\frac{\frac{30}{2} \pm \frac{2 \sqrt{335}}{2}}{\frac{20}{2}}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B%5Cfrac%7B30%7D%7B2%7D%20%5Cpm%20%5Cfrac%7B2%20%5Csqrt%7B335%7D%7D%7B2%7D%7D%7B%5Cfrac%7B20%7D%7B2%7D%7D)
![x=\frac{15 \pm \sqrt{335}}{10}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B15%20%5Cpm%20%5Csqrt%7B335%7D%7D%7B10%7D)
So we have these two x values:
![x=\frac{15+\sqrt{335}}{10} \text{ or } \frac{15-\sqrt{335}}{10}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B15%2B%5Csqrt%7B335%7D%7D%7B10%7D%20%5Ctext%7B%20or%20%7D%20%5Cfrac%7B15-%5Csqrt%7B335%7D%7D%7B10%7D)
Now we just need to find the corresponding y-coordinate for each pair of points.
I'm going to use the easier equation
.
Let's do it for the first x I mentioned:
If
then
.
Let's simplify:
Distribute the -3 to the terms on top:
![y=\frac{-45-3\sqrt{335}}{10}+5](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B-45-3%5Csqrt%7B335%7D%7D%7B10%7D%2B5)
Combine the two terms; I'm going to do this by writing 5 as 50/10:
![y=\frac{-45-3\sqrt{335}+50}{10}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B-45-3%5Csqrt%7B335%7D%2B50%7D%7B10%7D)
Combine like terms on top; the -45+50 part:
.
So one solution point is:
.
Let's find the other one for the other x that we got.
If
then
.
Let's simplify.
Distribute the -3 on top:
![y=\frac{-45+3\sqrt{335}}{10}+5](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B-45%2B3%5Csqrt%7B335%7D%7D%7B10%7D%2B5)
I'm going to write 5 as 50/10 so I can combine the terms as one fraction:
![y=\frac{-45+3\sqrt{335}+50}{5}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B-45%2B3%5Csqrt%7B335%7D%2B50%7D%7B5%7D)
Simplify the -45+50 part:
.
So the other point of intersection is:
.
The two solutions in exact form are:
![(\frac{15+\sqrt{335}}{10},\frac{5-3\sqrt{335}}{10})](https://tex.z-dn.net/?f=%28%5Cfrac%7B15%2B%5Csqrt%7B335%7D%7D%7B10%7D%2C%5Cfrac%7B5-3%5Csqrt%7B335%7D%7D%7B10%7D%29)
and
.
If you prefer to look at approximations just put into your calculator:
![(3.3303,-4.9909)](https://tex.z-dn.net/?f=%283.3303%2C-4.9909%29)
and
.