Question

select the correct answer.
what is the solution to this system of equations?
x + 3y - z = 8
4x + 2y = 8
2x + 5y - 2z = 14
a. (-1, 2, 1)
b. (-1, 2, -1)
c. (1, 2, 1)
d. (1, 2, -1)

Explanation:

Step1: Simplify the second equation

The second equation is \(4x + 2y = 8\). We can divide both sides by 2 to simplify it:
\(\frac{4x + 2y}{2}=\frac{8}{2}\)
\(2x + y = 4\)
Let's solve for \(y\): \(y = 4 - 2x\)

Step2: Analyze the first and third equations

The first equation is \(x + 3y - z = 8\), multiply it by 2: \(2x + 6y - 2z = 16\)
The third equation is \(2x + 5y - 2z = 14\)
Subtract the third equation from the multiplied first equation:
\((2x + 6y - 2z)-(2x + 5y - 2z)=16 - 14\)
Simplify: \(y = 2\)

Step3: Find the value of \(x\)

We know \(y = 2\) from Step 2. Substitute \(y = 2\) into \(y = 4 - 2x\):
\(2 = 4 - 2x\)
Solve for \(x\):
\(2x = 4 - 2\)
\(2x = 2\)
\(x = 1\)

Step4: Find the value of \(z\)

Substitute \(x = 1\) and \(y = 2\) into the first equation \(x + 3y - z = 8\):
\(1+3\times2 - z = 8\)
\(1 + 6 - z = 8\)
\(7 - z = 8\)
\(-z = 8 - 7\)
\(-z = 1\)
\(z=-1\)

Answer:

D. \((1, 2, -1)\)