Answer:
x = -2, y = -5
Step-by-step explanation:
Isolate x for 3x - 2y = 4
3x - 2y = 4
Add 2y to both sides
3x -2y + 2y = 4 + 2y
Simplify
3x = 4 + 2y
Divide both sides by 3
![\frac{3x}{3} =\frac{4}{3} +\frac{2y}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B3x%7D%7B3%7D%20%3D%5Cfrac%7B4%7D%7B3%7D%20%2B%5Cfrac%7B2y%7D%7B3%7D)
Simplify
![x=\frac{4+2y}{3}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B4%2B2y%7D%7B3%7D)
Now we substitute
for x in 4x - 3y = 7
![4*\frac{4+2y}{3}-3y=7](https://tex.z-dn.net/?f=4%2A%5Cfrac%7B4%2B2y%7D%7B3%7D-3y%3D7)
Isolate y for ![4*\frac{4+2y}{3}-3y=7](https://tex.z-dn.net/?f=4%2A%5Cfrac%7B4%2B2y%7D%7B3%7D-3y%3D7)
![4*\frac{4+2y}{3}-3y=7](https://tex.z-dn.net/?f=4%2A%5Cfrac%7B4%2B2y%7D%7B3%7D-3y%3D7)
Expand ![4*\frac{4+2y}{3}-3y](https://tex.z-dn.net/?f=4%2A%5Cfrac%7B4%2B2y%7D%7B3%7D-3y)
![4*\frac{4+2y}{3} =\frac{16+8y}{3}](https://tex.z-dn.net/?f=4%2A%5Cfrac%7B4%2B2y%7D%7B3%7D%20%3D%5Cfrac%7B16%2B8y%7D%7B3%7D)
![4*\frac{4+2y}{3}](https://tex.z-dn.net/?f=4%2A%5Cfrac%7B4%2B2y%7D%7B3%7D)
Multiply fractions: ![a*\frac{b}{c} =\frac{a*b}{c}](https://tex.z-dn.net/?f=a%2A%5Cfrac%7Bb%7D%7Bc%7D%20%3D%5Cfrac%7Ba%2Ab%7D%7Bc%7D)
![=\frac{(4+2y)*4}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%284%2B2y%29%2A4%7D%7B3%7D)
Expand ![(4+2y)*4](https://tex.z-dn.net/?f=%284%2B2y%29%2A4)
![=4(4+2y)](https://tex.z-dn.net/?f=%3D4%284%2B2y%29)
Apply the distributive law: ![a(b+c)=ab+ac](https://tex.z-dn.net/?f=a%28b%2Bc%29%3Dab%2Bac)
![a=4, b=4,c=2y](https://tex.z-dn.net/?f=a%3D4%2C%20b%3D4%2Cc%3D2y)
![=4*4+4*2y](https://tex.z-dn.net/?f=%3D4%2A4%2B4%2A2y)
Simplify ![4*4+4*2y](https://tex.z-dn.net/?f=4%2A4%2B4%2A2y)
Multiply the numbers: ![4*4 = 16](https://tex.z-dn.net/?f=4%2A4%20%3D%2016)
![=16+4*2y](https://tex.z-dn.net/?f=%3D16%2B4%2A2y)
Multiply the numbers: ![4*2=8](https://tex.z-dn.net/?f=4%2A2%3D8)
![=16+8y](https://tex.z-dn.net/?f=%3D16%2B8y)
![=\frac{16+8y}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B16%2B8y%7D%7B3%7D)
![=\frac{8y+16}{3}-3y](https://tex.z-dn.net/?f=%3D%5Cfrac%7B8y%2B16%7D%7B3%7D-3y)
Convert element to fraction: ![3y=\frac{3y3}{3}](https://tex.z-dn.net/?f=3y%3D%5Cfrac%7B3y3%7D%7B3%7D)
![=\frac{16+8y}{3} -\frac{3y*3}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B16%2B8y%7D%7B3%7D%20-%5Cfrac%7B3y%2A3%7D%7B3%7D)
Since the denominators are equal, combine the fractions:
±
![=\frac{16+8y-3y*3}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B16%2B8y-3y%2A3%7D%7B3%7D)
![16+8y-3y*3](https://tex.z-dn.net/?f=16%2B8y-3y%2A3)
Multiply the numbers: ![3*3=9](https://tex.z-dn.net/?f=3%2A3%3D9)
![=16+8y-9y](https://tex.z-dn.net/?f=%3D16%2B8y-9y)
Add similar elements: ![8y-9y=-y](https://tex.z-dn.net/?f=8y-9y%3D-y)
![=16-y](https://tex.z-dn.net/?f=%3D16-y)
![=\frac{16-y}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B16-y%7D%7B3%7D)
Apply the fraction rule:
±![\frac{b}{c}= \frac{a±b}{c}](https://tex.z-dn.net/?f=%5Cfrac%7Bb%7D%7Bc%7D%3D%20%5Cfrac%7Ba%C2%B1b%7D%7Bc%7D)
![\frac{16-y}{3} =\frac{16}{3} -\frac{y}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B16-y%7D%7B3%7D%20%3D%5Cfrac%7B16%7D%7B3%7D%20-%5Cfrac%7By%7D%7B3%7D)
![=\frac{16}{3}-\frac{y}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B16%7D%7B3%7D-%5Cfrac%7By%7D%7B3%7D)
![\frac{16}{3} -\frac{y}{3} =7](https://tex.z-dn.net/?f=%5Cfrac%7B16%7D%7B3%7D%20-%5Cfrac%7By%7D%7B3%7D%20%3D7)
Multiply both sides by 3
![\frac{16}{3}*3-\frac{y}{3} *3=7*3](https://tex.z-dn.net/?f=%5Cfrac%7B16%7D%7B3%7D%2A3-%5Cfrac%7By%7D%7B3%7D%20%2A3%3D7%2A3)
Simplify
![16-y=21](https://tex.z-dn.net/?f=16-y%3D21)
Subtract 16 from both sides
![16-y-16=21-16](https://tex.z-dn.net/?f=16-y-16%3D21-16)
Simplify
![-y=5](https://tex.z-dn.net/?f=-y%3D5)
Divide both sides by -1
![\frac{-y}{-1} =\frac{5}{-1}](https://tex.z-dn.net/?f=%5Cfrac%7B-y%7D%7B-1%7D%20%3D%5Cfrac%7B5%7D%7B-1%7D)
Simplify
![y=-5](https://tex.z-dn.net/?f=y%3D-5)
For
substitute ![y=-5](https://tex.z-dn.net/?f=y%3D-5)
![x=\frac{4+2y(-5)}{3}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B4%2B2y%28-5%29%7D%7B3%7D)
Remove parentheses: ![(-a)=-a](https://tex.z-dn.net/?f=%28-a%29%3D-a)
![=\frac{4+2y*5}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B4%2B2y%2A5%7D%7B3%7D)
Multiply the numbers: ![2*5=10](https://tex.z-dn.net/?f=2%2A5%3D10)
![=4-10](https://tex.z-dn.net/?f=%3D4-10)
Subtract the numbers: ![4-10=-6](https://tex.z-dn.net/?f=4-10%3D-6)
![=-6](https://tex.z-dn.net/?f=%3D-6)
![=\frac{-6}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B-6%7D%7B3%7D)
Apply the fraction rule: ![\frac{-a}{b} =-\frac{a}{b}](https://tex.z-dn.net/?f=%5Cfrac%7B-a%7D%7Bb%7D%20%3D-%5Cfrac%7Ba%7D%7Bb%7D)
![=-\frac{6}{3}](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B6%7D%7B3%7D)
Divide the numbers: ![\frac{6}{3}=2](https://tex.z-dn.net/?f=%5Cfrac%7B6%7D%7B3%7D%3D2)
![=-2](https://tex.z-dn.net/?f=%3D-2)
The solutions to the system of equations are
![y=-5,x=-2](https://tex.z-dn.net/?f=y%3D-5%2Cx%3D-2)
<u>Checking answers</u>
Plug in
and
into
and ![4x-3y=7](https://tex.z-dn.net/?f=4x-3y%3D7)
![3(-2)-2(-5)=4](https://tex.z-dn.net/?f=3%28-2%29-2%28-5%29%3D4)
Remove parentheses: ![(-a)=-a,-(-a)=a](https://tex.z-dn.net/?f=%28-a%29%3D-a%2C-%28-a%29%3Da)
![=3*2+2=5](https://tex.z-dn.net/?f=%3D3%2A2%2B2%3D5)
Multiply the numbers: ![3*2=6](https://tex.z-dn.net/?f=3%2A2%3D6)
![=-6+2*5](https://tex.z-dn.net/?f=%3D-6%2B2%2A5)
Multiply the numbers: ![2*5=10](https://tex.z-dn.net/?f=2%2A5%3D10)
![=-6+10](https://tex.z-dn.net/?f=%3D-6%2B10)
Add/subtract the numbers: ![-6+10=4](https://tex.z-dn.net/?f=-6%2B10%3D4)
![=4](https://tex.z-dn.net/?f=%3D4)
First equation proven true
Substitute the values of x and y into the second equation
![4(-2)-3(-5)=7](https://tex.z-dn.net/?f=4%28-2%29-3%28-5%29%3D7)
Follow the PEMDAS order of operations
Multiply and divide left to right ![4(-2)](https://tex.z-dn.net/?f=4%28-2%29)
![4(-2)](https://tex.z-dn.net/?f=4%28-2%29)
Apply rule ![a*(-b)=-a*b](https://tex.z-dn.net/?f=a%2A%28-b%29%3D-a%2Ab)
![4(-2)=-4*2=-8](https://tex.z-dn.net/?f=4%28-2%29%3D-4%2A2%3D-8)
![=-8](https://tex.z-dn.net/?f=%3D-8)
![=-8-3(-5)](https://tex.z-dn.net/?f=%3D-8-3%28-5%29)
Multiply and divide left to right ![3(-5)](https://tex.z-dn.net/?f=3%28-5%29)
![3(-5)](https://tex.z-dn.net/?f=3%28-5%29)
Apply rule ![a*(-b)=-a*b](https://tex.z-dn.net/?f=a%2A%28-b%29%3D-a%2Ab)
![3*(-5)=-3*5=-15](https://tex.z-dn.net/?f=3%2A%28-5%29%3D-3%2A5%3D-15)
![=-15](https://tex.z-dn.net/?f=%3D-15)
![=-8-(-15)](https://tex.z-dn.net/?f=%3D-8-%28-15%29)
Add and subtract left to right ![-8-(-15)](https://tex.z-dn.net/?f=-8-%28-15%29)
![-8-(-15)](https://tex.z-dn.net/?f=-8-%28-15%29)
Apply rule ![-(-a)=+a](https://tex.z-dn.net/?f=-%28-a%29%3D%2Ba)
![-(-15)=+15](https://tex.z-dn.net/?f=-%28-15%29%3D%2B15)
![=-8+15](https://tex.z-dn.net/?f=%3D-8%2B15)
![-8+15=7](https://tex.z-dn.net/?f=-8%2B15%3D7)
![=7](https://tex.z-dn.net/?f=%3D7)
Second equation proven true.
Both equations are true with
and