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].
