Question
What is the solution to the following system of equations? \( \left\{\begin{array}{l}y=3 x-9 \\ y=x+5\end{array}\right. \) (A) \( (1,6) \) (B) \( (2,12) \) (C) \( (3,-9) \) (D) \( (7,12) \)
Ask by Hall Riley. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is \( (7, 12) \), choice D.
Solution
To solve the system of equations:
\[
\begin{cases}
y = 3x - 9 \\
y = x + 5
\end{cases}
\]
**Step 1: Set the two equations equal to each other since both are equal to \( y \).**
\[
3x - 9 = x + 5
\]
**Step 2: Solve for \( x \).**
\[
3x - x = 5 + 9 \\
2x = 14 \\
x = 7
\]
**Step 3: Substitute \( x = 7 \) back into one of the original equations to find \( y \). Using the second equation:**
\[
y = 7 + 5 = 12
\]
**Solution: \( (7, 12) \)**
**Answer:** (D) \( (7,12) \)
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Simplify this solution The Deep Dive
To solve the system of equations \( y = 3x - 9 \) and \( y = x + 5 \), we can set the two expressions for \( y \) equal to each other: \[ 3x - 9 = x + 5. \] Now, let's solve for \( x \): 1. Subtract \( x \) from both sides: \[ 3x - x - 9 = 5 \] \[ 2x - 9 = 5. \] 2. Add 9 to both sides: \[ 2x = 14. \] 3. Divide by 2: \[ x = 7. \] Now, substitute \( x = 7 \) back into the second equation to find \( y \): \[ y = 7 + 5 = 12. \] Thus, the solution to the system is the point \( (7, 12) \), which corresponds to option (D).
