which statement correctly explains how mari could find the solution to the following system of linear…

Explanation:

Step1: Recall elimination method

The elimination method for a system of linear equations aims to eliminate one variable by making the coefficients of that variable (in absolute value) equal. Then we add or subtract the equations to eliminate the variable. For the system (\begin{cases}2f - 5g=-9\-7f + 3g = 4\end{cases}), we want to eliminate either (f) or (g). Let's check the coefficients of (f): (2) and (-7). To make their absolute values equal, we can multiply the first equation by (7) (the coefficient of (f) in the second equation, ignoring the sign) and the second equation by (2) (the coefficient of (f) in the first equation).

Step2: Analyze each option

• Option 1: Multiply first equation by (7): (7\times(2f - 5g)=7\times(-9)) gives (14f-35g = - 63). Multiply second equation by (2): (2\times(-7f + 3g)=2\times4) gives (-14f + 6g=8). Now, if we add these two new equations: ((14f-35g)+(-14f + 6g)=-63 + 8), which simplifies to (-29g=-55). Here, the (f) terms are eliminated ((14f-14f = 0)).
• Option 2: Multiply first equation by (3): (3\times(2f - 5g)=3\times(-9)) gives (6f-15g=-27). Multiply second equation by (5): (5\times(-7f + 3g)=5\times4) gives (-35f + 15g = 20). If we subtract, say first new equation - second new equation: ((6f-15g)-(-35f + 15g)=-27-20) which is (41f-30g=-47), no elimination. If we subtract the other way, it also won't eliminate a variable properly.
• Option 3: Multiply first equation by (-7): (-7\times(2f - 5g)=-7\times(-9)) gives (-14f + 35g = 63). Multiply second equation by (2): (2\times(-7f + 3g)=2\times4) gives (-14f+6g = 8). Adding these: ((-14f + 35g)+(-14f + 6g)=63 + 8) gives (-28f + 41g = 71), no elimination of (f) (coefficients of (f) are both (-14), subtraction would be needed, but the option says add).
• Option 4: Multiply first equation by (-3): (-3\times(2f - 5g)=-3\times(-9)) gives (-6f + 15g = 27). Multiply second equation by (-5): (-5\times(-7f + 3g)=-5\times4) gives (35f-15g=-20). Subtracting: ((-6f + 15g)-(35f - 15g)=27-(-20)) gives (-41f + 30g = 47), no elimination.

So, Option 1 is correct as it eliminates the (f) variable when we add the two new equations.

Answer:

A. Multiply the first equation by 7 and the second equation by 2, and then add.