Consider the system of linear equations.

Question

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 eliminate the \(x\)-terms in the system of equations using the linear combination method, we need to adjust the coefficients of \(x\) in both equations to make them equal in magnitude but opposite in sign.

The original equations are:

1. \(5x + 10y = 15\)
2. \(10x + 3y = 13\)

We can multiply the first equation by a factor that will make the coefficient of \(x\) in the first equation equal to \(-10\). The coefficient of \(x\) in the second equation is \(10\).

To achieve this, we need to multiply the first equation by \(-2\):

\[ -2(5x + 10y) = -2(15) \implies -10x - 20y = -30 \]

Now, our equations will be:

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

From this manipulation, we can see that multiplying the first equation by \(-2\) achieves the desired elimination of the \(x\)-terms.

Therefore, the answer is:

–2