Answer :
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
[tex]\frac{3x}{3} =\frac{4}{3} +\frac{2y}{3}[/tex]
Simplify
[tex]x=\frac{4+2y}{3}[/tex]
Now we substitute [tex]x=\frac{4+2y}{3}[/tex] for x in 4x - 3y = 7
[tex]4*\frac{4+2y}{3}-3y=7[/tex]
Isolate y for [tex]4*\frac{4+2y}{3}-3y=7[/tex]
[tex]4*\frac{4+2y}{3}-3y=7[/tex]
Expand [tex]4*\frac{4+2y}{3}-3y[/tex]
[tex]4*\frac{4+2y}{3} =\frac{16+8y}{3}[/tex]
[tex]4*\frac{4+2y}{3}[/tex]
Multiply fractions: [tex]a*\frac{b}{c} =\frac{a*b}{c}[/tex]
[tex]=\frac{(4+2y)*4}{3}[/tex]
Expand [tex](4+2y)*4[/tex]
[tex]=4(4+2y)[/tex]
Apply the distributive law: [tex]a(b+c)=ab+ac[/tex]
[tex]a=4, b=4,c=2y[/tex]
[tex]=4*4+4*2y[/tex]
Simplify [tex]4*4+4*2y[/tex]
Multiply the numbers: [tex]4*4 = 16[/tex]
[tex]=16+4*2y[/tex]
Multiply the numbers: [tex]4*2=8[/tex]
[tex]=16+8y[/tex]
[tex]=\frac{16+8y}{3}[/tex]
[tex]=\frac{8y+16}{3}-3y[/tex]
Convert element to fraction: [tex]3y=\frac{3y3}{3}[/tex]
[tex]=\frac{16+8y}{3} -\frac{3y*3}{3}[/tex]
Since the denominators are equal, combine the fractions: [tex]\frac{a}{c}[/tex]±[tex]\frac{b}{c}= \frac{a±b}{c}[/tex]
[tex]=\frac{16+8y-3y*3}{3}[/tex]
[tex]16+8y-3y*3[/tex]
Multiply the numbers: [tex]3*3=9[/tex]
[tex]=16+8y-9y[/tex]
Add similar elements: [tex]8y-9y=-y[/tex]
[tex]=16-y[/tex]
[tex]=\frac{16-y}{3}[/tex]
Apply the fraction rule: [tex]\frac{a}{c}[/tex]±[tex]\frac{b}{c}= \frac{a±b}{c}[/tex]
[tex]\frac{16-y}{3} =\frac{16}{3} -\frac{y}{3}[/tex]
[tex]=\frac{16}{3}-\frac{y}{3}[/tex]
[tex]\frac{16}{3} -\frac{y}{3} =7[/tex]
Multiply both sides by 3
[tex]\frac{16}{3}*3-\frac{y}{3} *3=7*3[/tex]
Simplify
[tex]16-y=21[/tex]
Subtract 16 from both sides
[tex]16-y-16=21-16[/tex]
Simplify
[tex]-y=5[/tex]
Divide both sides by -1
[tex]\frac{-y}{-1} =\frac{5}{-1}[/tex]
Simplify
[tex]y=-5[/tex]
For [tex]x=\frac{4+2y}{3}[/tex] substitute [tex]y=-5[/tex]
[tex]x=\frac{4+2y(-5)}{3}[/tex]
Remove parentheses: [tex](-a)=-a[/tex]
[tex]=\frac{4+2y*5}{3}[/tex]
Multiply the numbers: [tex]2*5=10[/tex]
[tex]=4-10[/tex]
Subtract the numbers: [tex]4-10=-6[/tex]
[tex]=-6[/tex]
[tex]=\frac{-6}{3}[/tex]
Apply the fraction rule: [tex]\frac{-a}{b} =-\frac{a}{b}[/tex]
[tex]=-\frac{6}{3}[/tex]
Divide the numbers: [tex]\frac{6}{3}=2[/tex]
[tex]=-2[/tex]
The solutions to the system of equations are
[tex]y=-5,x=-2[/tex]
Checking answers
Plug in [tex]x=-2[/tex] and [tex]y=-5[/tex] into [tex]3x-2y=4[/tex] and [tex]4x-3y=7[/tex]
[tex]3(-2)-2(-5)=4[/tex]
Remove parentheses: [tex](-a)=-a,-(-a)=a[/tex]
[tex]=3*2+2=5[/tex]
Multiply the numbers: [tex]3*2=6[/tex]
[tex]=-6+2*5[/tex]
Multiply the numbers: [tex]2*5=10[/tex]
[tex]=-6+10[/tex]
Add/subtract the numbers: [tex]-6+10=4[/tex]
[tex]=4[/tex]
First equation proven true
Substitute the values of x and y into the second equation
[tex]4(-2)-3(-5)=7[/tex]
Follow the PEMDAS order of operations
Multiply and divide left to right [tex]4(-2)[/tex]
[tex]4(-2)[/tex]
Apply rule [tex]a*(-b)=-a*b[/tex]
[tex]4(-2)=-4*2=-8[/tex]
[tex]=-8[/tex]
[tex]=-8-3(-5)[/tex]
Multiply and divide left to right [tex]3(-5)[/tex]
[tex]3(-5)[/tex]
Apply rule [tex]a*(-b)=-a*b[/tex]
[tex]3*(-5)=-3*5=-15[/tex]
[tex]=-15[/tex]
[tex]=-8-(-15)[/tex]
Add and subtract left to right [tex]-8-(-15)[/tex]
[tex]-8-(-15)[/tex]
Apply rule [tex]-(-a)=+a[/tex]
[tex]-(-15)=+15[/tex]
[tex]=-8+15[/tex]
[tex]-8+15=7[/tex]
[tex]=7[/tex]
Second equation proven true.
Both equations are true with [tex]x=-2[/tex] and [tex]y=-5[/tex]