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10) \( \begin{array}{c}-4 x+9 y=9 \\ x-3 y=-6\end{array} \)

Ask by Fernandez Salazar. in the United States
Feb 25,2025

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The solution is \( x = 9 \) and \( y = 5 \).

Solución

Solve the system of equations \( -4x+9y=9;x-3y=-6 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}-4x+9y=9\\x-3y=-6\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}-4x+9y=9\\x=-6+3y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(-4\left(-6+3y\right)+9y=9\) - step3: Simplify: \(24-3y=9\) - step4: Move the constant to the right side: \(-3y=9-24\) - step5: Subtract the numbers: \(-3y=-15\) - step6: Change the signs: \(3y=15\) - step7: Divide both sides: \(\frac{3y}{3}=\frac{15}{3}\) - step8: Divide the numbers: \(y=5\) - step9: Substitute the value of \(y:\) \(x=-6+3\times 5\) - step10: Calculate: \(x=9\) - step11: Calculate: \(\left\{ \begin{array}{l}x=9\\y=5\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=9\\y=5\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(9,5\right)\) The solution to the system of equations is \( (x, y) = (9, 5) \).

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Beyond the Answer

To solve the system of equations given by \( -4x + 9y = 9 \) and \( x - 3y = -6 \), let's first isolate \( x \) in the second equation. Rearranging gives us \( x = 3y - 6 \). Now, substitute this expression for \( x \) into the first equation: \[ -4(3y - 6) + 9y = 9. \] Distributing the \(-4\) yields: \[ -12y + 24 + 9y = 9. \] Now combine like terms: \[ -3y + 24 = 9. \] Subtract 24 from both sides: \[ -3y = -15. \] Dividing by \(-3\) results in: \[ y = 5. \] Now, substitute \( y = 5 \) back into the equation for \( x \): \[ x = 3(5) - 6 = 15 - 6 = 9. \] Thus, the solution to the system of equations is \( x = 9 \) and \( y = 5 \). In summary, the final solution is \( (x, y) = (9, 5) \).

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