Question

The point (5, -3) is a solution to which system of equations?

y= - 1/2x +6
y = 3x - 1

y = 2/3x + 6
3y + 6x - 18 = 0

y = x + 2
y = x + 5

y = x - 8
2x + y = 7

Answer 1

To determine which system of equations the point (5, -3) is a solution to, we need to substitute \(x = 5\) and \(y = -3\) into each of the given pairs of equations and check if the point satisfies both equations in any system.

1. **System 1:**
- \(y = -\frac{1}{2}x + 6\)
- Substitute \(x = 5\):
\[
y = -\frac{1}{2}(5) + 6 = -\frac{5}{2} + 6 = -\frac{5}{2} + \frac{12}{2} = \frac{7}{2} \quad \text{(not -3)}
\]
- This equation is not satisfied.
- Now for \(y = 3x - 1\):
\[
y = 3(5) - 1 = 15 - 1 = 14 \quad \text{(not -3)}
\]
- This system is not satisfied.

2. **System 2:**
- \(y = \frac{2}{3}x + 6\)
- Substitute \(x = 5\):
\[
y = \frac{2}{3}(5) + 6 = \frac{10}{3} + 6 = \frac{10}{3} + \frac{18}{3} = \frac{28}{3} \quad \text{(not -3)}
\]
- Now for \(3y + 6x - 18 = 0\):
\[
3(-3) + 6(5) - 18 = -9 + 30 - 18 = 3 \quad \text{(not 0)}
\]
- This system is not satisfied.

3. **System 3:**
- \(y = x + 2\)
- Substitute \(x = 5\):
\[
y = 5 + 2 = 7 \quad \text{(not -3)}
\]
- Now for \(y = x + 5\):
\[
y = 5 + 5 = 10 \quad \text{(not -3)}
\]
- This system is not satisfied.

4. **System 4:**
- \(y = x - 8\)
- Substitute \(x = 5\):
\[
y = 5 - 8 = -3 \quad \text{(satisfied)}
\]
- Now for \(2x + y = 7\):
\[
2(5) + (-3) = 10 - 3 = 7 \quad \text{(satisfied)}
\]
- This system is satisfied.

Thus, the point (5, -3) is a solution to the fourth system of equations:

\[
\begin{align*}
y &= x - 8 \\
2x + y &= 7
\end{align*}
\]