Questions: Question 4 Solve the system of equations by elimination. 2x - 6y = -72 9x - 7y = -144 x = y =

Transcript text: Question 4 Solve the system of equations by elimination. \[ \begin{array}{l} 2 x-6 y=-72 \\ 9 x-7 y=-144 \end{array} \] \[ x= \] $y=$

Solution

Solution Steps

To solve the system of equations using elimination, we aim to eliminate one of the variables by making the coefficients of either $x$ or $y$ the same in both equations. We can multiply the first equation by 9 and the second equation by 2 to align the coefficients of $x$. Then, we subtract one equation from the other to eliminate $x$ and solve for $y$. Once $y$ is found, substitute it back into one of the original equations to solve for $x$.

Step 1: Align Coefficients for Elimination

To eliminate one of the variables, we align the coefficients of $x$ by multiplying the first equation by 9 and the second equation by 2: \[ \begin{align_} 9(2x - 6y) &= 9(-72) \\ 2(9x - 7y) &= 2(-144) \end{align_} \] This results in: \[ \begin{align_} 18x - 54y &= -648 \\ 18x - 14y &= -288 \end{align_} \]

Step 2: Eliminate $x$

Subtract the second equation from the first to eliminate $x$: \[ (18x - 54y) - (18x - 14y) = -648 - (-288) \] \[ -54y + 14y = -648 + 288 \] \[ -40y = -360 \]

Step 3: Solve for $y$

Divide both sides by $-40$ to solve for $y$: \[ y = \frac{-360}{-40} = 9 \]

Step 4: Substitute $y$ to Solve for $x$

Substitute $y = 9$ back into the first original equation: \[ 2x - 6(9) = -72 \] \[ 2x - 54 = -72 \] \[ 2x = -72 + 54 \] \[ 2x = -18 \] \[ x = \frac{-18}{2} = -9 \]