Answer:
C. [tex](x=8,y=-5,z=0)[/tex]
Step-by-step explanation:
We have been given a system of equations and we are asked to solve our given system of equations.
[tex]3x+4y+4z=4...(1)[/tex]
[tex]5x+7y+3z=5...(2)[/tex]
[tex]4x+5y+7z=7...(3)[/tex]
Using equation (1) and equation (2) we will eliminate z.
[tex]3*3x+4*3y+4*3z=4*3...(1)[/tex]
[tex]5*4x+7*4y+3*4z=5*4...(2)[/tex]
[tex]9x+12y+12z=12...(1)[/tex]
[tex]20x+28y+12z=20...(2)[/tex]
Subtracting equation (1) from equation (2) we will get,
[tex]11x+16y=8...(4)[/tex]
Similarly we will eliminate z using equation (2) and (3).
[tex]5*7x+7*7y+3*7z=5*7...(2)[/tex]
[tex]4*3x+5*3y+7*3z=7*3...(3)[/tex]
[tex]35x+49y+21z=35...(2)[/tex]
[tex]12x+15y+21z=21...(3)[/tex]
Subtracting equation (3) from equation (2) we will get,
[tex]23x+34y=14...(5)[/tex]
Now we will use substitution method to solve equation (4) and (5).
From equation (4) we will get,
[tex]x=\frac{8-16y}{11}[/tex]
Substituting this value in equation (5) we will get,
[tex]23(\frac{8-16y}{11})+34y=14[/tex]
[tex]\frac{184-368y}{11}+34y=14[/tex]
Let us have a common denominator.
[tex]\frac{184-368y}{11}+\frac{34*11y}{11}=14[/tex]
[tex]\frac{184-368y}{11}+\frac{374y}{11}=14[/tex]
[tex]\frac{184-368y+374y}{11}=14[/tex]
[tex]\frac{184+6y}{11}=14[/tex]
[tex]\frac{184+6y}{11}*11=14*11[/tex]
[tex]184+6y=154[/tex]
[tex]184-184+6y=154-184[/tex]
[tex]6y=-30[/tex]
[tex]\frac{6y}{6}=\frac{-30}{6}[/tex]
[tex]y=-5[/tex]
Therefore, y equals negative 5.
Substituting [tex]y=-5[/tex] in equation (4) we will get,
[tex]11x+16(-5)=8[/tex]
[tex]11x-80=8[/tex]
[tex]11x-80+80=8+80[/tex]
[tex]11x=88[/tex]
[tex]\frac{11x}{11}=\frac{88}{11}[/tex]
[tex]x=8[/tex]
Therefore, x equals 8.
Substituting [tex]y=-5[/tex] and [tex]x=8[/tex] in equation (1) we will get,
[tex]3*8+4(-5)+4z=4[/tex]
[tex]24-20+4z=4[/tex]
[tex]4+4z=4[/tex]
[tex]4-4+4z=4-4[/tex]
[tex]4z=0[/tex]
[tex]z=0[/tex]
Therefore, the value of z is 0.
Upon looking at our given choices we can see that option C is the correct choice.
