Question

A system of equations is shown.
3x−2y=23
x−3y=3
Which system of equations has the same solution as the system of equations shown?

2x−5y=7
3x+y=13

2x+5y=7
3x−y=13

5x−3y=39
2x−y=16

5x+3y=39
2x+y=16

Answer 1

To determine which system of equations has the same solution as the given system:

1. **Original system of equations:**
\[
\begin{align*}
3x - 2y &= 23 \quad \text{(1)} \\
x - 3y &= 3 \quad \text{(2)}
\end{align*}
\]

Let's solve this system to find the solution (x, y).

**From equation (2):**
\[
x = 3 + 3y
\]
Substituting \(x\) from equation (2) into equation (1):
\[
3(3 + 3y) - 2y = 23
\]
Expanding this gives:
\[
9 + 9y - 2y = 23
\]
Combining like terms:
\[
9 + 7y = 23
\]
Subtracting 9 from both sides:
\[
7y = 14
\]
Dividing by 7:
\[
y = 2
\]

**Now substituting \(y = 2\) back to find \(x\):**
\[
x = 3 + 3(2) = 3 + 6 = 9
\]
Thus, the solution to the original system is \( (x, y) = (9, 2) \).

Next, we need to check which of the given systems has the same solution \( (9, 2) \):

1. **First option:**
\[
\begin{align*}
2x - 5y &= 7 \quad \text{(3)} \\
3x + y &= 13 \quad \text{(4)}
\end{align*}
\]

**Checking (3):**
\[
2(9) - 5(2) = 18 - 10 = 8 \quad \text{(not true)}
\]
So, this option is eliminated.

2. **Second option:**
\[
\begin{align*}
2x + 5y &= 7 \quad \text{(5)} \\
3x - y &= 13 \quad \text{(6)}
\end{align*}
\]

**Checking (5):**
\[
2(9) + 5(2) = 18 + 10 = 28 \quad \text{(not true)}
\]
So, this option is also eliminated.

3. **Third option:**
\[
\begin{align*}
5x - 3y &= 39 \quad \text{(7)} \\
2x - y &= 16 \quad \text{(8)}
\end{align*}
\]

**Checking (7):**
\[
5(9) - 3(2) = 45 - 6 = 39 \quad \text{(true)}
\]

**Checking (8):**
\[
2(9) - 2 = 18 - 2 = 16 \quad \text{(true)}
\]
Both equations are satisfied, so this option is valid.

4. **Fourth option:**
\[
\begin{align*}
5x + 3y &= 39 \quad \text{(9)} \\
2x + y &= 16 \quad \text{(10)}
\end{align*}
\]

**Checking (9):**
\[
5(9) + 3(2) = 45 + 6 = 51 \quad \text{(not true)}
\]
So this option is eliminated as well.

After checking all options, the system of equations with the same solution as the original system is:
\[
\boxed{5x - 3y = 39 \quad \text{and} \quad 2x - y = 16}
\]