Question

solve the following system using the elimination method. enter your answer as an ordered pair in the form (x,y) if there is one, unique solution. enter all if there are infinitely many solutions and enter none if there are no solutions. 6x - 5y = 37 2x + 6y = -26

Explanation:

Step1: Eliminate x by multiplying the second equation

Multiply the second equation \(2x + 6y=-26\) by 3 to get \(6x + 18y=-78\).
$$3\times(2x + 6y)=3\times(-26)\Rightarrow 6x + 18y=-78$$

Step2: Subtract the first equation from the new equation

Subtract \(6x - 5y = 37\) from \(6x + 18y=-78\):
$$(6x + 18y)-(6x - 5y)=-78 - 37$$
Simplify the left side: \(6x+18y - 6x + 5y = 23y\)
Simplify the right side: \(-115\)
So, \(23y=-115\)

Step3: Solve for y

Divide both sides by 23:
$$y=\frac{-115}{23}=-5$$

Step4: Substitute y into one of the original equations

Substitute \(y = - 5\) into \(2x+6y=-26\):
$$2x+6\times(-5)=-26$$
Simplify: \(2x-30=-26\)

Step5: Solve for x

Add 30 to both sides: \(2x=-26 + 30=4\)
Divide by 2: \(x = 2\)

Answer:

\((2, -5)\)