To solve for [tex]\( y \)[/tex] by eliminating the [tex]\( x \)[/tex] terms in the given system of equations:
First equation: [tex]\( 6x - 5y = 17 \)[/tex]
Second equation: [tex]\( 7x + 3y = 11 \)[/tex]
We need to make the coefficients of the [tex]\( x \)[/tex] terms equal in magnitude but opposite in sign. Here’s a step-by-step solution to achieve that:
1. Identify the coefficients of [tex]\( x \)[/tex]:
- In the first equation, the coefficient of [tex]\( x \)[/tex] is 6.
- In the second equation, the coefficient of [tex]\( x \)[/tex] is 7.
2. Determine how to equalize the coefficients:
- To eliminate [tex]\( x \)[/tex], we can make the coefficients of [tex]\( x \)[/tex] in the two equations equal in magnitude but opposite in sign.
- We will use the least common multiple (LCM) of the coefficients of [tex]\( x \)[/tex], which is 42. This means we want the coefficients of [tex]\( x \)[/tex] to become [tex]\( +42 \)[/tex] and [tex]\( -42 \)[/tex].
3. Decide on the multipliers to reach the target coefficients:
- Multiply the first equation by 7:
[tex]\[
6 \cdot 7 = 42
\][/tex]
- Multiply the second equation by -6:
[tex]\[
7 \cdot (-6) = -42
\][/tex]
4. Apply the chosen multipliers to each equation:
- The first equation multiplied by 7 becomes:
[tex]\[
7 \cdot (6x - 5y) = 7 \cdot 17
\][/tex]
[tex]\[
42x - 35y = 119
\][/tex]
- The second equation multiplied by -6 becomes:
[tex]\[
-6 \cdot (7x + 3y) = -6 \cdot 11
\][/tex]
[tex]\[
-42x - 18y = -66
\][/tex]
5. Add the two resulting equations to eliminate [tex]\( x \)[/tex]:
[tex]\[
(42x - 35y) + (-42x - 18y) = 119 + (-66)
\][/tex]
[tex]\[
0x - 53y = 53
\][/tex]
[tex]\[
-53y = 53
\][/tex]
Thus, the correct multipliers to eliminate [tex]\( x \)[/tex] and solve for [tex]\( y \)[/tex] are 7 for the first equation and -6 for the second equation.
Therefore, the answer is:
The first equation should be multiplied by 7 and the second equation by -6.
