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.

To determine the constants by which the equations should be multiplied to eliminate the $ x $ terms, let's follow these steps:

Given equations:
1. $ 6x - 5y = 17 $ (First equation)
2. $ 7x + 3y = 11 $ (Second equation)

We want to eliminate the $ x $-terms by making their coefficients equal and opposite. To achieve this:

1. The coefficient of $ x $ in the first equation is $ 6 $.
2. The coefficient of $ x $ in the second equation is $ 7 $.

To make these coefficients equal and opposite, we need to find numbers that can be multiplied by $ 6 $ and $ 7 $ to make their products equal in magnitude but opposite in signs:

- The least common multiple (LCM) of $ 6 $ and $ 7 $ is $ 42 $.

To get coefficients of $ 42 $ and $ -42 $:

1. Multiply the first equation by $ 7 $:
[tex]\[ 7 \cdot (6x - 5y) = 7 \cdot 17 \][/tex]
[tex]\[ 42x - 35y = 119 \][/tex]

2. Multiply the second equation by $ -6 $:
[tex]\[ -6 \cdot (7x + 3y) = -6 \cdot 11 \][/tex]
[tex]\[ -42x - 18y = -66 \][/tex]

Now, the coefficients of $ x $ in the two modified equations are $ 42 $ and $ -42 $, which will cancel each other out when the equations are added.

Hence, the correct constants to multiply the equations by are:
- Multiply the first equation by $ 7 $
- Multiply the second equation by $ -6 $

Thus, the correct choice is:
The first equation should be multiplied by [tex]$ 7 $[/tex] and the second equation by [tex]$ -6 $[/tex].