To solve the given system of linear equations using the linear combination method to eliminate the [tex]\( x \)[/tex]-terms, let's outline the process step-by-step:
The given system of linear equations is:
[tex]\[
\begin{cases}
5x + 10y = 15 \\
10x + 3y = 13
\end{cases}
\][/tex]
To eliminate the [tex]\( x \)[/tex]-terms, we aim to make the coefficients of [tex]\( x \)[/tex] in both equations equal in magnitude but opposite in sign. This way, adding the two equations will cancel out the [tex]\( x \)[/tex]-terms.
1. Identify the coefficients of [tex]\( x \)[/tex] in both equations.
- For the first equation [tex]\( 5x + 10y = 15 \)[/tex], the coefficient of [tex]\( x \)[/tex] is 5.
- For the second equation [tex]\( 10x + 3y = 13 \)[/tex], the coefficient of [tex]\( x \)[/tex] is 10.
2. Determine the least common multiple (LCM) of the coefficients of [tex]\( x \)[/tex] in both equations, which is 10 in this case.
3. Adjust the first equation so that its [tex]\( x \)[/tex]-coefficient matches in magnitude but is opposite in sign to the [tex]\( x \)[/tex]-coefficient of the second equation. To do this, we need a coefficient of [tex]\( x \)[/tex] in the first equation to be [tex]\(-10\)[/tex] (opposite in sign to 10).
4. Calculate the multiplication factor for the first equation to achieve this:
- Since the [tex]\( x \)[/tex]-coefficient in the first equation is 5, to get [tex]\(-10\)[/tex], multiply the entire first equation by [tex]\(-2\)[/tex].
Thus, the first equation should be multiplied by:
[tex]\[
-2
\][/tex]
This will transform the first equation as follows:
[tex]\[
-2 \cdot (5x + 10y) = -2 \cdot 15
\][/tex]
Which simplifies to:
[tex]\[
-10x - 20y = -30
\][/tex]
By multiplying the first equation by [tex]\(-2\)[/tex], we now have the same magnitude but opposite sign for the coefficients of [tex]\( x \)[/tex] in both equations, allowing us to eliminate the [tex]\( x \)[/tex]-terms through addition.
So, the correct answer is:
[tex]\[
-2
\][/tex]
