which constants can be multiplied by the equations so one variable will be eliminated when the systems are…

Explanation:

Step1: Recall elimination - method concept

To eliminate a variable when adding two linear equations (a_1x + b_1y=c_1) and (a_2x + b_2y=c_2), we make the coefficients of the variable to be eliminated equal in magnitude and opposite in sign. For variable (x) with coefficients (a_1 = 5) and (a_2=12) in the equations (5x + 13y=232) and (12x + 7y=218) respectively, if we multiply the first - equation by (- 12) and the second by (5), the (x) - terms will be (-12\times5x=-60x) and (5\times12x = 60x). When added, (-60x+60x = 0). For variable (y) with coefficients (b_1 = 13) and (b_2 = 7), if we want to eliminate (y), we should multiply the first equation by (-7) and the second by (13) (or vice - versa with opposite signs) to get (-7\times13y=-91y) and (13\times7y = 91y).

Step2: Check the options

• Option 1: If we multiply the first equation by (-13) and the second by (7), the (y) - terms are (-13\times13y=-169y) and (7\times7y = 49y), (y) is not eliminated.
• Option 2: If we multiply the first equation by (7) and the second by (13), the (y) - terms are (7\times13y = 91y) and (13\times7y = 91y), (y) is not eliminated.
• Option 3: If we multiply the first equation by (-12) and the second by (5), the (x) - terms are (-12\times5x=-60x) and (5\times12x = 60x). When the two new equations are added, the (x) variable is eliminated.
• Option 4: If we multiply the first equation by (5) and the second by (12), the (x) - terms are (5\times5x = 25x) and (12\times12x=144x), (x) is not eliminated.

Answer:

The first equation can be multiplied by (-12) and the second equation by (5) to eliminate (x).