Solve the following system of equations:

[tex]\[
\left\{
\begin{array}{l}
5x - 6y + z = 49 \\
-x + 5y + 3z = -14 \\
2x - 2y + 2z = 28
\end{array}
\right.
\][/tex]

Show your work here:



Answer :

To solve this system of linear equations step-by-step, we aim to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations:

[tex]\[ \left\{\begin{array}{l} 5x - 6y + z = 49 \quad \text{(1)} \\ -x + 5y + 3z = -14 \quad \text{(2)} \\ 2x - 2y + 2z = 28 \quad \text{(3)} \end{array}\right. \][/tex]

Step 1: Simplify Equation (3)

Divide all terms in equation (3) by 2:
[tex]\[ x - y + z = 14 \quad \text{(4)} \][/tex]

Now our system looks like:
[tex]\[ \left\{\begin{array}{l} 5x - 6y + z = 49 \quad \text{(1)} \\ -x + 5y + 3z = -14 \quad \text{(2)} \\ x - y + z = 14 \quad \text{(4)} \end{array}\right. \][/tex]

Step 2: Eliminate one variable

First, let's eliminate [tex]\( z \)[/tex] between equations (1) and (4) by subtracting (4) from (1).

[tex]\[ (5x - 6y + z) - (x - y + z) = 49 - 14 \][/tex]

Simplify the left side:
[tex]\[ 4x - 5y = 35 \quad \text{(5)} \][/tex]

Next, eliminate [tex]\( z \)[/tex] between equations (2) and (4). Multiply equation (4) by 3 to line up like terms with equation (2):

[tex]\[ 3(x - y + z) = 3(14) \implies 3x - 3y + 3z = 42 \][/tex]

Subtract this equation from equation (2):
[tex]\[ (-x + 5y + 3z) - (3x - 3y + 3z) = -14 - 42 \][/tex]

Simplify the left side:
[tex]\[ -4x + 8y = -56 \quad \text{(6)} \][/tex]

Step 3: Simplify and Solve the New System

We now have a simpler system with two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \left\{\begin{array}{l} 4x - 5y = 35 \quad \text{(5)} \\ -4x + 8y = -56 \quad \text{(6)} \end{array}\right. \][/tex]

Add equations (5) and (6) to eliminate [tex]\( x \)[/tex]:

[tex]\[ (4x - 5y) + (-4x + 8y) = 35 - 56 \][/tex]

This simplifies to:
[tex]\[ 3y = -21 \][/tex]

Solving for [tex]\( y \)[/tex]:
[tex]\[ y = -7 \][/tex]

Step 4: Substitute [tex]\( y \)[/tex] back into one of the simplified equations

Using equation (5):
[tex]\[ 4x - 5(-7) = 35 \implies 4x + 35 = 35 \][/tex]

Solving for [tex]\( x \)[/tex]:
[tex]\[ 4x = 0 \implies x = 0 \][/tex]

Step 5: Substitute [tex]\( x \)[/tex] and [tex]\( y \)[/tex] back into one of the original equations

Use equation (4) to solve for [tex]\( z \)[/tex]:
[tex]\[ 0 - (-7) + z = 14 \implies 7 + z = 14 \][/tex]

Solving for [tex]\( z \)[/tex]:
[tex]\[ z = 7 \][/tex]

Final Answer:

The solution to the system of equations is:
[tex]\[ x = 0, \quad y = -7, \quad z = 7 \][/tex]

Thus, the values that satisfy the original system of equations are [tex]\( x = 0 \)[/tex], [tex]\( y = -7 \)[/tex], and [tex]\( z = 7 \)[/tex].