Solve the following system of equations with the substitution method:
\[
\left\{\begin{array}{lll}
-3 x+12 y & =-24 \\
x & =4 y+8
\end{array}\right.
\]
Answer
Write the final answer: The system of equations has infinitely many solutions, and any pair \((x, y)\) that satisfies \(x = 4y + 8\) is a solution. So, the final answer is \(\boxed{\text{The system of equations has infinitely many solutions}}\)
Step 1 ：Understand the problem: We are given a system of two equations and we are asked to solve it using the substitution method.
Step 2 ：Start the solution: The substitution method involves solving one of the equations for one variable in terms of the other variable and then substituting this expression into the other equation.
Step 3 ：Solve one of the equations for one variable: The second equation is already solved for x, so we can use it as it is. We have: \(x = 4y + 8\)
Step 4 ：Substitute the expression for x into the other equation: Substitute \(x = 4y + 8\) into the first equation: \(-3(4y + 8) + 12y = -24\) which simplifies to \(-12y - 24 + 12y = -24\) and further simplifies to \(-24 = -24\)
Step 5 ：Solve for y: The y terms cancel out, leaving \(-24 = -24\). This is a true statement, which means that the system of equations has infinitely many solutions.
Step 6 ：Check the solution: Since the y terms cancel out and we are left with a true statement, this means that any value of y will satisfy the first equation as long as \(x = 4y + 8\).
Step 7 ：Write the final answer: The system of equations has infinitely many solutions, and any pair \((x, y)\) that satisfies \(x = 4y + 8\) is a solution. So, the final answer is \(\boxed{\text{The system of equations has infinitely many solutions}}\)
