Question

Consider the system of linear equations.

5 x + 10 y = 15. 10 x + 3 y = 13

To use the linear combination method and addition to eliminate the x-terms, by which number should the first equation be multiplied?
–2
Negative one-half
One-half
2

Answer 1

To use the linear combination method (also known as the elimination method) to eliminate the \( x \)-terms in the given system of equations:

1. The first equation is:
\( 5x + 10y = 15 \) (Equation 1)
2. The second equation is:
\( 10x + 3y = 13 \) (Equation 2)

To eliminate \( x \), we can make the coefficients of \( x \) in both equations equal. Currently, the coefficient of \( x \) in Equation 1 is \( 5 \) and in Equation 2 it is \( 10 \).

To make the coefficient of \( x \) in the first equation equal to \( 10 \), we can multiply the entire first equation by \( 2 \).

So, we multiply Equation 1 by \( 2 \):

\[
2(5x + 10y) = 2(15)
\]

This gives:

\[
10x + 20y = 30 \quad (Equation 3)
\]

Now we have:

1. \( 10x + 20y = 30 \) (Equation 3)
2. \( 10x + 3y = 13 \) (Equation 2)

Since both equations have the same \( x \)-coefficient (10), we can subtract Equation 2 from Equation 3 to eliminate \( x \).

Therefore, the number by which the first equation should be multiplied to use the elimination method is \( 2 \).