Question

select all of the ways this system of equations can be solved.
6x - y = 9
3x + 4y = 18
a) multiply the first equation by 2, then subtract the equations.
b) multiply the second equation by -2, then add the equations.
c) multiply the first equation by -4, then subtract the equations.
d) multiply the second equation by 2, then add the equations.
e) multiply the first equation by 4, then add the equations.

Explanation:

To solve a system of linear equations using elimination, we aim to eliminate one variable by making the coefficients of that variable opposites (for addition) or the same (for subtraction). Let's analyze each option:

Step 1: Analyze Option A

First equation: \(6x - y = 9\), multiply by 2: \(12x - 2y = 18\)
Second equation: \(3x + 4y = 18\)
Subtracting: \((12x - 2y) - (3x + 4y) = 18 - 18\) → \(9x - 6y = 0\). We didn't eliminate a variable. So A is incorrect.

Step 2: Analyze Option B

Second equation: \(3x + 4y = 18\), multiply by -2: \(-6x - 8y = -36\)
First equation: \(6x - y = 9\)
Adding: \((6x - y) + (-6x - 8y) = 9 + (-36)\) → \(-9y = -27\). We eliminated \(x\). So B is correct.

Step 3: Analyze Option C

First equation: \(6x - y = 9\), multiply by -4: \(-24x + 4y = -36\)
Second equation: \(3x + 4y = 18\)
Subtracting: \((-24x + 4y) - (3x + 4y) = -36 - 18\) → \(-27x = -54\). We eliminated \(y\). So C is correct.

Step 4: Analyze Option D

Second equation: \(3x + 4y = 18\), multiply by 2: \(6x + 8y = 36\)
First equation: \(6x - y = 9\)
Adding: \((6x - y) + (6x + 8y) = 9 + 36\) → \(12x + 7y = 45\). We didn't eliminate a variable. So D is incorrect.

Step 5: Analyze Option E

First equation: \(6x - y = 9\), multiply by 4: \(24x - 4y = 36\)
Second equation: \(3x + 4y = 18\)
Adding: \((24x - 4y) + (3x + 4y) = 36 + 18\) → \(27x = 54\). We eliminated \(y\). So E is correct.

Answer:

B. Multiply the second equation by -2, then add the equations.
C. Multiply the first equation by -4, then subtract the equations.
E. Multiply the first equation by 4, then add the equations.