Let's approach this step by step to understand which constants should be used to multiply each equation to eliminate the [tex]\( x \)[/tex] terms.
We start with the two given equations:
1) [tex]\( 6x - 5y = 17 \)[/tex]
2) [tex]\( 7x + 3y = 11 \)[/tex]
To eliminate the [tex]\( x \)[/tex] terms, we need to make the coefficients of [tex]\( x \)[/tex] in both equations equal (but opposite in sign). The coefficients of [tex]\( x \)[/tex] in the equations are 6 and 7, respectively.
To make these coefficients equal, we can multiply the first equation by the coefficient of [tex]\( x \)[/tex] in the second equation, and vice versa. Specifically, we will multiply:
- The first equation by 7
- The second equation by -6
Thus, the equations become:
1) [tex]\( 7 \times (6x - 5y) = 7 \times 17 \)[/tex]
which simplifies to:
[tex]\( 42x - 35y = 119 \)[/tex]
2) [tex]\( -6 \times (7x + 3y) = -6 \times 11 \)[/tex]
which simplifies to:
[tex]\( -42x - 18y = -66 \)[/tex]
Now we see that the coefficients of [tex]\( x \)[/tex] (42 and -42) are equal in magnitude but opposite in sign, allowing us to eliminate the [tex]\( x \)[/tex] terms by adding the two equations together.
So the constants to multiply the equations by are:
- Multiply the first equation by 7
- Multiply the second equation by -6
Thus, the correct answer is:
The first equation should be multiplied by 7 and the second equation by -6.
