To solve the system of equations:
[tex]\[
\begin{cases}
6x - 2y = -2 \\
y = 3x + 12
\end{cases}
\][/tex]
we can use the substitution method. Here are the detailed steps:
1. Substitute the value of [tex]\( y \)[/tex] from the second equation into the first equation:
[tex]\[ y = 3x + 12 \][/tex]
Substitute [tex]\( y \)[/tex] in the first equation:
[tex]\[ 6x - 2(3x + 12) = -2 \][/tex]
2. Simplify the equation:
Distribute the [tex]\( -2 \)[/tex]:
[tex]\[ 6x - 6x - 24 = -2 \][/tex]
Combine like terms:
[tex]\[ 0 - 24 = -2 \][/tex]
This simplifies to:
[tex]\[ -24 = -2 \][/tex]
3. Analyze the result:
The simplified equation [tex]\( -24 = -2 \)[/tex] is a contradiction; it is not true. This means that there is no solution that satisfies both equations simultaneously.
Therefore, the system of equations:
[tex]\[
\begin{cases}
6x - 2y = -2 \\
y = 3x + 12
\end{cases}
\][/tex]
has no solutions. This indicates that the lines represented by these equations are parallel and do not intersect.
