Question

Solve using elimination.

–x − 5y = –19
9x + y = –5

Answer 1

To solve the system of equations using the elimination method, we'll start with the given equations:

1. \(-x - 5y = -19\)
2. \(9x + y = -5\)

First, we can rearrange the first equation to express it in a more standard form:

\[
-x - 5y + 19 = 0 \quad \Rightarrow \quad x + 5y = 19
\]

Now we can use the two equations:

1. \(x + 5y = 19\) (Equation 1)
2. \(9x + y = -5\) (Equation 2)

Next, we'll manipulate these equations to eliminate one of the variables. Let’s eliminate \(y\) by multiplying Equation 2 by 5:

\[
5(9x + y) = 5(-5) \quad \Rightarrow \quad 45x + 5y = -25 \quad \text{(Equation 3)}
\]

Now we have:

1. \(x + 5y = 19\) (Equation 1)
2. \(45x + 5y = -25\) (Equation 3)

Next, we can subtract Equation 1 from Equation 3 to eliminate \(y\):

\[
(45x + 5y) - (x + 5y) = -25 - 19
\]

This simplifies to:

\[
(45x - x) + (5y - 5y) = -44
\]
\[
44x = -44
\]

Now divide both sides by 44:

\[
x = -1
\]

Now that we have the value of \(x\), we can substitute it back into one of the original equations to find \(y\). We can use Equation 1:

\[
-x - 5y = -19
\]
Substituting \(x = -1\):

\[
-(-1) - 5y = -19
\]
\[
1 - 5y = -19
\]
Subtract \(1\) from both sides:

\[
-5y = -19 - 1
\]
\[
-5y = -20
\]

Now divide both sides by \(-5\):

\[
y = 4
\]

Thus, the solution to the system of equations is:

\[
x = -1, \quad y = 4
\]

So the final answer is:

\[
\boxed{(-1, 4)}
\]