Question

for each ordered pair, determine whether it is a solution to the system of equations.
\

$$\begin{cases} 2x - 3y = -8 \\\\ 9x + 2y = -5 \\end{cases}$$

$(x, y)$	is it a solution?
$(-2, 1)$	$\circ$	$\circ$
$(1, -7)$	$\circ$	$\circ$
$(5, 6)$	$\circ$	$\circ$
$(0, -4)$	$\circ$	$\circ$

Explanation:

To determine if an ordered pair \((x, y)\) is a solution to the system of equations, we substitute the values of \(x\) and \(y\) into both equations and check if both equations are satisfied.

For \((-2, 1)\):

Step 1: Substitute into \(2x - 3y = -8\)

Substitute \(x = -2\) and \(y = 1\) into the first equation:
\(2(-2) - 3(1) = -4 - 3 = -7\)
Since \(-7 eq -8\), this ordered pair does not satisfy the first equation. So, we can already conclude it's not a solution. But let's check the second equation for completeness.

Step 2: Substitute into \(9x + 2y = -5\)

Substitute \(x = -2\) and \(y = 1\) into the second equation:
\(9(-2) + 2(1) = -18 + 2 = -16\)
Since \(-16 eq -5\), it also doesn't satisfy the second equation.

For \((1, -7)\):

Step 1: Substitute into \(2x - 3y = -8\)

Substitute \(x = 1\) and \(y = -7\) into the first equation:
\(2(1) - 3(-7) = 2 + 21 = 23\)
Since \(23 eq -8\), this ordered pair does not satisfy the first equation.

Step 2: Substitute into \(9x + 2y = -5\)

Substitute \(x = 1\) and \(y = -7\) into the second equation:
\(9(1) + 2(-7) = 9 - 14 = -5\)
It satisfies the second equation but not the first, so it's not a solution.

For \((5, 6)\):

Step 1: Substitute into \(2x - 3y = -8\)

Substitute \(x = 5\) and \(y = 6\) into the first equation:
\(2(5) - 3(6) = 10 - 18 = -8\)
This satisfies the first equation. Now check the second equation.

Step 2: Substitute into \(9x + 2y = -5\)

Substitute \(x = 5\) and \(y = 6\) into the second equation:
\(9(5) + 2(6) = 45 + 12 = 57\)
Since \(57 eq -5\), it does not satisfy the second equation. So, it's not a solution.

For \((0, -4)\):

Step 1: Substitute into \(2x - 3y = -8\)

Substitute \(x = 0\) and \(y = -4\) into the first equation:
\(2(0) - 3(-4) = 0 + 12 = 12\)
Since \(12 eq -8\), this ordered pair does not satisfy the first equation.

Step 2: Substitute into \(9x + 2y = -5\)

Substitute \(x = 0\) and \(y = -4\) into the second equation:
\(9(0) + 2(-4) = 0 - 8 = -8\)
Since \(-8 eq -5\), it also doesn't satisfy the second equation.

Wait, but let's check again. Wait, maybe I made a mistake. Wait, let's re - check the pair \((-2,1)\) for the first equation: \(2x-3y\), \(x = - 2\), \(y = 1\): \(2\times(-2)-3\times1=-4 - 3=-7 eq - 8\). For the second equation: \(9\times(-2)+2\times1=-18 + 2=-16 eq - 5\).

Wait, maybe there is a mistake in my calculation. Wait, let's solve the system of equations properly to find the correct solution.

We have the system:
\(\begin{cases}2x-3y=-8&(1)\\9x + 2y=-5&(2)\end{cases}\)

Let's use the elimination method. Multiply equation \((1)\) by \(2\) and equation \((2)\) by \(3\) to eliminate \(y\):

Equation \((1)\times2\): \(4x-6y=-16\) \((3)\)

Equation \((2)\times3\): \(27x+6y=-15\) \((4)\)

Add equations \((3)\) and \((4)\):

\((4x + 27x)+(-6y + 6y)=-16-15\)

\(31x=-31\)

\(x=-1\)

Now substitute \(x = - 1\) into equation \((1)\):

\(2\times(-1)-3y=-8\)

\(-2-3y=-8\)

\(-3y=-8 + 2=-6\)

\(y = 2\)

So the solution to the system is \((-1,2)\). So none of the given ordered pairs are solutions? Wait, but let's re - check the ordered pairs:

1. For \((-2,1)\):
- First equation: \(2\times(-2)-3\times1=-4 - 3=-7 eq - 8\)
- Second equation: \(9\times(-2)+2\times1=-18 + 2=-16 eq - 5\) → No

2. For \((1,-7)\):
- First equation: \(2\times1-3\times(-7)=2 + 21=23 eq - 8\)
- Second equation: \(9\times1+2\times(-7)=9-14=-5\). Wait, the second equation is satisfied, but the first is not. So overall, it's not a solution.

3. For \((5,6)\):
- First equation: \(2\times5-3\times6=10 - 18=-8\) (satisfies the first equation)
- Second equation: \(9\times5+2\times6=45 + 12=57 eq - 5\) (does not satisfy the second equation) → No

4. For \((0,-4)\):
- First equation: \(2\times0-3\times(-4)=0 + 12=12 eq - 8\)
- Second equation: \(9\times0+2\times(-4)=0-8=-8 eq - 5\) → No

So the answers are:

For \((-2,1)\): No

For \((1,-7)\): No

For \((5,6)\): No

For \((0,-4)\): No

Answer:

• For \((-2,1)\): No
• For \((1,-7)\): No
• For \((5,6)\): No
• For \((0,-4)\): No