To find the value of [tex]\( y \)[/tex] in the given system of equations, let's follow these steps:
The system of equations is:
[tex]\[
\begin{cases}
2x - 6y = 12 \\
-6x + 14y = 42
\end{cases}
\][/tex]
### Step 1: Simplify the equations if possible
First, check if the equations can be simplified:
For the first equation:
[tex]\[ 2x - 6y = 12 \][/tex]
Divide both sides by 2:
[tex]\[ x - 3y = 6 \][/tex]
For the second equation:
[tex]\[ -6x + 14y = 42 \][/tex]
Divide both sides by 2 for simplicity:
[tex]\[ -3x + 7y = 21 \][/tex]
So, the simplified system of equations is:
[tex]\[
\begin{cases}
x - 3y = 6 \\
-3x + 7y = 21
\end{cases}
\][/tex]
### Step 2: Solve one equation for one variable
Solve the first equation for [tex]\( x \)[/tex]:
[tex]\[ x - 3y = 6 \][/tex]
[tex]\[ x = 6 + 3y \][/tex]
### Step 3: Substitute this expression into the second equation
Substitute [tex]\( x = 6 + 3y \)[/tex] into the second equation:
[tex]\[ -3(6 + 3y) + 7y = 21 \][/tex]
Expand and simplify:
[tex]\[ -18 - 9y + 7y = 21 \][/tex]
Combine like terms:
[tex]\[ -18 - 2y = 21 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Add 18 to both sides:
[tex]\[ -2y = 39 \][/tex]
Divide both sides by -2:
[tex]\[ y = -\frac{39}{2} \][/tex]
Thus, the value of [tex]\( y \)[/tex] is:
[tex]\[ y = -\frac{39}{2} \][/tex]
