Question

Solving systems by elimination 15x+y=-13 -5x+9y=23

Answer 1

To solve the system of equations using the elimination method, we have the following equations:

1) \( 15x + y = -13 \) (Equation 1)
2) \( -5x + 9y = 23 \) (Equation 2)

### Step 1: Eliminate one variable

We can eliminate \( y \) by manipulating the equations. To do this, we can multiply Equation 1 by \( -9 \) so that the coefficients of \( y \) have the same magnitude but opposite signs.

Multiply Equation 1 by \( -9 \):

\[
-9(15x + y) = -9(-13)
\]

This simplifies to:

\[
-135x - 9y = 117 \quad \text{(Equation 3)}
\]

Now we have:

3) \( -135x - 9y = 117 \)

Next, we will keep Equation 2 as is:

2) \( -5x + 9y = 23 \)

### Step 2: Add the two equations together

Now, we can add Equations 2 and 3 together to eliminate \( y \):

\[
(-135x - 9y) + (-5x + 9y) = 117 + 23
\]

This simplifies to:

\[
-140x = 140
\]

### Step 3: Solve for \( x \)

Divide both sides by \( -140 \):

\[
x = \frac{140}{-140} = -1
\]

### Step 4: Substitute back to find \( y \)

Now substitute \( x = -1 \) back into Equation 1 to find \( y \):

\[
15(-1) + y = -13
\]

This simplifies to:

\[
-15 + y = -13
\]

Adding 15 to both sides gives:

\[
y = 2
\]

### Conclusion

The solution to the system of equations is:

\[
x = -1, \quad y = 2
\]

So, the solution is \(\boxed{(-1, 2)}\).