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To eliminate the [tex]\( x \)[/tex] terms and solve for [tex]\( y \)[/tex] in the fewest steps, by which constants should the equations be multiplied before adding the equations together?

First equation: [tex]\( 6x - 5y = 17 \)[/tex]
Second equation: [tex]\( 7x + 3y = 11 \)[/tex]

A. The first equation should be multiplied by 3 and the second equation by -5.
B. The first equation should be multiplied by 3 and the second equation by 5.
C. The first equation should be multiplied by 7 and the second equation by -6.
D. The first equation should be multiplied by 7 and the second equation by 6.

Sagot :

GinnyAnswer
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.
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