To determine which option will allow us to eliminate one variable when adding the given system of equations, let's analyze each option in detail:
Given system of equations:
1. [tex]\(4x - 2y = 7\)[/tex]
2. [tex]\(3x - 3y = 15\)[/tex]
Option A: Multiply the top equation by -3 and the bottom equation by 2
1. Multiply the top equation by -3:
[tex]\(-3(4x - 2y) = -3(7)\)[/tex]
[tex]\(-12x + 6y = -21\)[/tex]
2. Multiply the bottom equation by 2:
[tex]\(2(3x - 3y) = 2(15)\)[/tex]
[tex]\(6x - 6y = 30\)[/tex]
3. Add the scaled equations together:
[tex]\(-12x + 6y + 6x - 6y = -21 + 30\)[/tex]
[tex]\(-6x = 9\)[/tex]
Since one variable (y) is eliminated, this option works. Therefore, Option A is valid for eliminating one variable.
Option B: Multiply the top equation by 3 and the bottom equation by 2
1. Multiply the top equation by 3:
[tex]\(3(4x - 2y) = 3(7)\)[/tex]
[tex]\(12x - 6y = 21\)[/tex]
2. Multiply the bottom equation by 2:
[tex]\(2(3x - 3y) = 2(15)\)[/tex]
[tex]\(6x - 6y = 30\)[/tex]
3. Add the scaled equations together:
[tex]\(12x - 6y + 6x - 6y = 21 + 30\)[/tex]
[tex]\(18x - 12y = 51\)[/tex]
Since neither variable is eliminated, this option does not work.
Option C: Multiply the top equation by [tex]\( \frac{1}{3} \)[/tex]
1. Multiply the top equation by [tex]\( \frac{1}{3} \)[/tex]:
[tex]\( \frac{1}{3}(4x - 2y) = \frac{1}{3}(7)\)[/tex]
[tex]\(\frac{4}{3}x - \frac{2}{3}y = \frac{7}{3}\)[/tex]
2. The bottom equation remains the same:
[tex]\(3x - 3y = 15\)[/tex]
3. Add the scaled equations together:
[tex]\(\frac{4}{3}x - \frac{2}{3}y + 3x - 3y = \frac{7}{3} + 15\)[/tex]
Converting to a common denominator, this does not eliminate any variable straightforwardly.
Option D: Multiply the top equation by 3 and the bottom equation by 4
1. Multiply the top equation by 3:
[tex]\(3(4x - 2y) = 3(7)\)[/tex]
[tex]\(12x - 6y = 21\)[/tex]
2. Multiply the bottom equation by 4:
[tex]\(4(3x - 3y) = 4(15)\)[/tex]
[tex]\(12x - 12y = 60\)[/tex]
3. Add the scaled equations together:
[tex]\(12x - 6y + 12x - 12y = 21 + 60\)[/tex]
[tex]\(24x - 18y = 81\)[/tex]
Since neither variable is eliminated, this option does not work.
Based on the detailed analysis, the correct answer is:
A. Multiply the top equation by -3 and the bottom equation by 2
