Question

Solve using substitution.
-10x - 2y = 2
5x + 2y = 8
(_,_)

Answer 1

To solve the system of equations using substitution, we start with the given equations:

1. \(-10x - 2y = 2\) (Equation 1)
2. \(5x + 2y = 8\) (Equation 2)

First, we can solve one of the equations for one variable in terms of the other. Let's solve Equation 2 for \(y\):

\[
5x + 2y = 8
\]

Subtract \(5x\) from both sides:

\[
2y = 8 - 5x
\]

Now, divide by 2:

\[
y = 4 - \frac{5}{2}x
\]

Now we have \(y\) expressed in terms of \(x\). We can substitute this expression for \(y\) back into Equation 1.

Substituting into Equation 1:

\[
-10x - 2(4 - \frac{5}{2}x) = 2
\]

Distributing the \(-2\):

\[
-10x - 8 + 5x = 2
\]

Now combine like terms:

\[
-10x + 5x - 8 = 2
\]

This simplifies to:

\[
-5x - 8 = 2
\]

Now, add 8 to both sides:

\[
-5x = 2 + 8
\]

\[
-5x = 10
\]

Now, divide by \(-5\):

\[
x = -2
\]

Now that we have \(x\), we can now substitute it back into the equation we found for \(y\):

\[
y = 4 - \frac{5}{2}(-2)
\]

Calculating:

\[
y = 4 + \frac{5}{2} \cdot 2
\]

\[
y = 4 + 5 = 9
\]

The solution to the system of equations is:

\[
(x, y) = (-2, 9)
\]

So the final answer is:

\[
\boxed{(-2, 9)}
\]