Question

Which systems of equations has the solution -2, 1?

Answer 1

To find systems of equations that have the solution \((-2, 1)\), we can create two equations that both hold true when \(x = -2\) and \(y = 1\). Here are a couple of examples:

### Example 1:
1. First equation:
\[
3x + 2y = 0
\]
Plugging in \((-2, 1)\):
\[
3(-2) + 2(1) = -6 + 2 = -4 \quad \text{(not true, so let's adjust this equation)}
\]

To make it true, let’s use:
\[
x + 2y = 0
\]
Check:
\[
-2 + 2(1) = -2 + 2 = 0 \quad \text{(true)}
\]

2. Second equation:
\[
y - x = 3
\]
Check:
\[
1 - (-2) = 1 + 2 = 3 \quad \text{(true)}
\]

Our system of equations can thus be:
\[
\begin{cases}
x + 2y = 0 \\
y - x = 3
\end{cases}
\]

### Example 2:
1. First equation:
\[
2x + y = -3
\]
Check:
\[
2(-2) + 1 = -4 + 1 = -3 \quad \text{(true)}
\]

2. Second equation:
\[
4x - 2y = -10
\]
Check:
\[
4(-2) - 2(1) = -8 - 2 = -10 \quad \text{(true)}
\]

Our second system of equations can be:
\[
\begin{cases}
2x + y = -3 \\
4x - 2y = -10
\end{cases}
\]

Any system of equations that results in the point \((-2, 1)\) as a solution will work. You can create many such systems by adjusting coefficients and constants in similar linear forms, ensuring that they both maintain the same solution.