To solve the system of equations:
[tex]\[
\begin{array}{l}
6x - 3y = 3 \\
-2x + 6y = 14
\end{array}
\][/tex]
we need to eliminate either the [tex]\(x\)[/tex]-terms or the [tex]\(y\)[/tex]-terms by adding the equations together. Let's determine the necessary multipliers for each case.
### Eliminating the [tex]\(x\)[/tex]-terms:
To eliminate the [tex]\(x\)[/tex]-terms, we want the coefficients of [tex]\(x\)[/tex] in both equations to have the same magnitude but opposite signs. Looking at the coefficients:
- The coefficient of [tex]\(x\)[/tex] in the first equation is [tex]\(6\)[/tex].
- The coefficient of [tex]\(x\)[/tex] in the second equation is [tex]\(-2\)[/tex].
We need to make the magnitude of the coefficients equal, which can be achieved by finding a common multiple. In this case, the least common multiple of 6 and 2 is 6.
To make the [tex]\(x\)[/tex]-terms cancel, we multiply the second equation by [tex]\(3\)[/tex] (since [tex]\(-2 \times 3 = -6\)[/tex]):
[tex]\[ -2x \times 3 = -6x \][/tex]
This makes the coefficients of [tex]\(x\)[/tex] in both equations [tex]\(6\)[/tex] and [tex]\(-6\)[/tex], respectively, allowing them to cancel out when the equations are added together.
So, the number we would multiply the second equation by in order to eliminate the [tex]\(x\)[/tex]-terms is:
[tex]\[
\boxed{3}
\][/tex]
### Eliminating the [tex]\(y\)[/tex]-terms:
To eliminate the [tex]\(y\)[/tex]-terms, we want the coefficients of [tex]\(y\)[/tex] in both equations to have the same magnitude but opposite signs. Looking at the coefficients:
- The coefficient of [tex]\(y\)[/tex] in the first equation is [tex]\(-3\)[/tex].
- The coefficient of [tex]\(y\)[/tex] in the second equation is [tex]\(6\)[/tex].
We need to make the magnitude of the coefficients equal, which can be achieved by finding a common multiple. In this case, the least common multiple of 3 and 6 is 6.
To make the [tex]\(y\)[/tex]-terms cancel, we multiply the first equation by [tex]\(2\)[/tex] (since [tex]\(-3 \times 2 = -6\)[/tex]):
[tex]\[ -3y \times 2 = -6y \][/tex]
This makes the coefficients of [tex]\(y\)[/tex] in both equations [tex]\(-6\)[/tex] and [tex]\(6\)[/tex], respectively, allowing them to cancel out when the equations are added together.
So, the number we would multiply the first equation by in order to eliminate the [tex]\(y\)[/tex]-terms is:
[tex]\[
\boxed{2}
\][/tex]
Thus, we have determined that to solve the system by elimination, you would multiply the second equation by [tex]\(3\)[/tex] to eliminate the [tex]\(x\)[/tex]-terms and multiply the first equation by [tex]\(2\)[/tex] to eliminate the [tex]\(y\)[/tex]-terms.
