Question

Using the linear combination method, what is the solution to the system of linear equations 5 x + 3 y = negative 10 and Negative 20 x minus 7 y = 15?
(–5, 1)
(–1, 5)
(1, –5)
(5, –1)

Answer 1

To solve the system of equations using the linear combination (elimination) method, we start with the two equations given:

1. \( 5x + 3y = -10 \) (Equation 1)
2. \( -20x - 7y = 15 \) (Equation 2)

First, we can simplify Equation 2 by multiplying it by -1 to make calculations easier:

\[
20x + 7y = -15 \quad \text{(Equation 2 simplified)}
\]

Now we have:

1. \( 5x + 3y = -10 \)
2. \( 20x + 7y = -15 \)

Next, we will eliminate \( x \). To do this, we can multiply Equation 1 by 4:

\[
4(5x + 3y) = 4(-10)
\]
\[
20x + 12y = -40 \quad \text{(Equation 3)}
\]

Now we have:

1. \( 20x + 12y = -40 \) (Equation 3)
2. \( 20x + 7y = -15 \) (Equation 2 simplified)

Now, we can eliminate \( x \) by subtracting Equation 2 from Equation 3:

\[
(20x + 12y) - (20x + 7y) = -40 - (-15)
\]
\[
12y - 7y = -40 + 15
\]
\[
5y = -25
\]

Dividing both sides by 5:

\[
y = -5
\]

Now that we have \( y \), we can substitute \( y \) back into Equation 1 to find \( x \):

\[
5x + 3(-5) = -10
\]
\[
5x - 15 = -10
\]
\[
5x = -10 + 15
\]
\[
5x = 5
\]
\[
x = 1
\]

Thus, the solution to the system of equations is:

\[
(x, y) = (1, -5)
\]

So the correct answer is:

**(1, –5)**.