To eliminate the [tex]\( x \)[/tex] terms and solve for [tex]\( y \)[/tex] in the fewest steps, we start with the given system of equations:
First equation: [tex]\(6x - 5y = 17\)[/tex]
Second equation: [tex]\(7x + 3y = 11\)[/tex]
To eliminate the [tex]\( x \)[/tex]-terms, we need to find constants by which to multiply each equation so that the coefficients of [tex]\( x \)[/tex] in both equations are opposites of each other.
We start by determining the least common multiple (LCM) of the coefficients of [tex]\( x \)[/tex], which are 6 and 7. The LCM of 6 and 7 is 42. We want the [tex]\( x \)[/tex]-terms to cancel out when the equations are added.
1. Multiply the first equation by 7:
[tex]\[
7(6x - 5y) = 7(17)
\][/tex]
This gives:
[tex]\[
42x - 35y = 119
\][/tex]
2. Multiply the second equation by -6:
[tex]\[
-6(7x + 3y) = -6(11)
\][/tex]
This gives:
[tex]\[
-42x - 18y = -66
\][/tex]
Now, we have the transformed equations:
[tex]\[
42x - 35y = 119
\][/tex]
[tex]\[
-42x - 18y = -66
\][/tex]
Adding these two equations together:
[tex]\[
(42x - 42x) + (-35y - 18y) = 119 + (-66)
\][/tex]
This simplifies to:
[tex]\[
0x - 53y = 53
\][/tex]
[tex]\[
-53y = 53
\][/tex]
[tex]\[
y = -1
\][/tex]
Thus, to eliminate the [tex]\( x \)[/tex]-terms by adding the equations together, the first equation should be multiplied by 7 and the second equation by -6.
Therefore, the correct answer is:
The first equation should be multiplied by 7 and the second equation by -6.
