To eliminate the variable [tex]\( y \)[/tex] from the given system of equations, we can follow these steps:
Given the system of equations:
[tex]\[
\left\{\begin{array}{r}
x + y - z = -3 \quad \text{(Equation 1)} \\
7x - 6y - 3z = -35 \quad \text{(Equation 2)} \\
2x + y - 4z = -14 \quad \text{(Equation 3)}
\end{array}\right.
\][/tex]
### Step 1: Eliminate [tex]\( y \)[/tex] from Equations 1 and 2
1. Multiply Equation 1 by 6:
[tex]\[
6(x + y - z) = 6(-3)
\][/tex]
This gives:
[tex]\[
6x + 6y - 6z = -18 \quad \text{(Equation 4)}
\][/tex]
2. Subtract Equation 2 from Equation 4:
[tex]\[
(6x + 6y - 6z) - (7x - 6y - 3z) = -18 - (-35)
\][/tex]
Simplifying this:
[tex]\[
6x + 6y - 6z - 7x + 6y + 3z = -18 + 35
\][/tex]
Combining like terms:
[tex]\[
-x + 12y - 3z = 17
\][/tex]
Multiplying the entire equation by [tex]\(-1\)[/tex] to simplify:
[tex]\[
x - 12y + 3z = -17 \quad \text{(Resulting Equation)}
\][/tex]
### Step 2: Eliminate [tex]\( y \)[/tex] from Equations 1 and 3
1. Multiply Equation 1 by 1:
[tex]\[
1(x + y - z) = 1(-3)
\][/tex]
This gives:
[tex]\[
x + y - z = -3 \quad \text{(Equation 5)}
\][/tex]
2. Subtract Equation 3 from Equation 5:
[tex]\[
(x + y - z) - (2x + y - 4z) = -3 - (-14)
\][/tex]
Simplifying this:
[tex]\[
x + y - z - 2x - y + 4z = -3 + 14
\][/tex]
Combining like terms:
[tex]\[
-x + 3z = 11
\][/tex]
Multiplying the entire equation by [tex]\(-1\)[/tex] to simplify:
[tex]\[
x - 3z = -11 \quad \text{(Resulting Equation)}
\][/tex]
### Summary:
- To eliminate [tex]\( y \)[/tex] from Equations 1 and 2, multiply Equation 1 by [tex]\( 6 \)[/tex].
- To eliminate [tex]\( y \)[/tex] from Equations 1 and 3, multiply Equation 1 by [tex]\( 1 \)[/tex].
The resulting equations are:
[tex]\[
\left\{
\begin{array}{l}
x - 12y + 3z = -17 \\
x - 3z = -11
\end{array}
\right.
\][/tex]
