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11) \( \begin{aligned}-3 x+7 y & =-16 \\ -9 x+5 y & =16\end{aligned} \)

Ask by Ray Rose. in the United States
Feb 25,2025

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Answer

\( x = -4 \) and \( y = -4 \).

Solution

Solve the system of equations \( -3x+7y=-16;-9x+5y=16 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}-3x+7y=-16\\-9x+5y=16\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=\frac{16+7y}{3}\\-9x+5y=16\end{array}\right.\) - step2: Substitute the value of \(x:\) \(-9\times \frac{16+7y}{3}+5y=16\) - step3: Simplify: \(-48-16y=16\) - step4: Move the constant to the right side: \(-16y=16+48\) - step5: Add the numbers: \(-16y=64\) - step6: Change the signs: \(16y=-64\) - step7: Divide both sides: \(\frac{16y}{16}=\frac{-64}{16}\) - step8: Divide the numbers: \(y=-4\) - step9: Substitute the value of \(y:\) \(x=\frac{16+7\left(-4\right)}{3}\) - step10: Calculate: \(x=-4\) - step11: Calculate: \(\left\{ \begin{array}{l}x=-4\\y=-4\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=-4\\y=-4\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(-4,-4\right)\) The solution to the system of equations is \( x = -4 \) and \( y = -4 \).

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To solve the system of equations \(-3x + 7y = -16\) and \(-9x + 5y = 16\), we can use either substitution or elimination. Let's choose the elimination method! First, we can multiply the first equation by 3 to align the coefficients of \(x\): \[ \begin{aligned} -9x + 21y &= -48 \quad (1)\\ -9x + 5y &= 16 \quad (2) \end{aligned} \] Now, subtract equation (2) from equation (1): \[ (21y - 5y) = -48 - 16 \] \[ 16y = -64 \] \[ y = -4 \] Now that we have \(y\), let's substitute it back into one of the original equations to find \(x\). Using the first equation: \[ -3x + 7(-4) = -16 \] \[ -3x - 28 = -16 \] \[ -3x = 12 \] \[ x = -4 \] So, the solution to the system is \(x = -4\) and \(y = -4\). The solution set is \((x, y) = (-4, -4)\).

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