To solve the given system of equations using substitution, we proceed as follows:
[tex]\[
\left\{
\begin{array}{l}
-4 x + y = 5 \\
3 y = 12 + 12 x
\end{array}
\right.
\][/tex]
Step 1: Solve one of the equations for one variable in terms of the other. We'll start with the first equation:
[tex]\[ -4 x + y = 5 \][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[ y = 4 x + 5 \][/tex]
Step 2: Substitute this value of [tex]\( y \)[/tex] into the second equation:
[tex]\[ 3 y = 12 + 12 x \][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( 4 x + 5 \)[/tex]:
[tex]\[ 3 (4 x + 5) = 12 + 12 x \][/tex]
Step 3: Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 12 x + 15 = 12 + 12 x \][/tex]
Subtract [tex]\( 12 x \)[/tex] from both sides of the equation:
[tex]\[ 15 = 12 \][/tex]
Step 4: Analyze the result:
The equation [tex]\( 15 = 12 \)[/tex] is a contradiction; it is never true. This means the system of equations has no solution.
Conclusion:
The given system of equations has no solution.
We conclude that there is no pair [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously. Therefore, the correct answer is "No solution".
