Question
Solve the system by the substitution method. \[ \begin{array}{r}x-2 y=-3 \\ y=-4 x+24\end{array} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \{\square \). (Type an ordered pair.) B. There are infinitely many solutions. C. There is no solution.
Ask by Sanders Lewis. in the United States
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is \( (5, 4) \).
Solution
To solve the system using the substitution method, follow these steps:
1. **Express one variable in terms of the other:**
From the second equation:
\[
y = -4x + 24
\]
2. **Substitute this expression into the first equation:**
\[
x - 2(-4x + 24) = -3
\]
Simplify:
\[
x + 8x - 48 = -3
\]
\[
9x - 48 = -3
\]
3. **Solve for \( x \):**
\[
9x = 45
\]
\[
x = 5
\]
4. **Substitute \( x = 5 \) back into the expression for \( y \):**
\[
y = -4(5) + 24 = -20 + 24 = 4
\]
5. **Write the solution as an ordered pair:**
\[
(5, 4)
\]
**Answer:**
A. The solution set is \( \{\, (5,\ 4) \,\} \).
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Simplify this solution Extra Insights
To solve the system by substitution, we start with the equation \( y = -4x + 24 \) and substitute this expression for \( y \) in the first equation \( x - 2y = -3 \). Thus, we have: \[ x - 2(-4x + 24) = -3 \] Expanding this gives: \[ x + 8x - 48 = -3 \] \[ 9x - 48 = -3 \] Now, adding 48 to both sides: \[ 9x = 45 \] Dividing by 9: \[ x = 5 \] Next, we substitute \( x = 5 \) back into the equation for \( y \): \[ y = -4(5) + 24 \] \[ y = -20 + 24 = 4 \] Thus, the solution set is \( \{(5, 4)\} \). Choose option A: A. The solution set is \( \{(5, 4)\} \).
