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Question 6(Multiple Choice Worth 1 points) \( \begin{array}{l}(05.02 \text { MC) } \\ \text { Solve the following system of equations: } \\ -2 x+3 y=7 \\ y=3 x+7\end{array} \)

Ask by Huang Nichols. in the United States
Jan 24,2025

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\( x = -2 \) and \( y = 1 \).

Solución

Solve the system of equations \( -2x+3y=7;y=3x+7 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}-2x+3y=7\\y=3x+7\end{array}\right.\) - step1: Substitute the value of \(y:\) \(-2x+3\left(3x+7\right)=7\) - step2: Simplify: \(7x+21=7\) - step3: Move the constant to the right side: \(7x=7-21\) - step4: Subtract the numbers: \(7x=-14\) - step5: Divide both sides: \(\frac{7x}{7}=\frac{-14}{7}\) - step6: Divide the numbers: \(x=-2\) - step7: Substitute the value of \(x:\) \(y=3\left(-2\right)+7\) - step8: Simplify: \(y=1\) - step9: Calculate: \(\left\{ \begin{array}{l}x=-2\\y=1\end{array}\right.\) - step10: Check the solution: \(\left\{ \begin{array}{l}x=-2\\y=1\end{array}\right.\) - step11: Rewrite: \(\left(x,y\right) = \left(-2,1\right)\) The solution to the system of equations is \( x = -2 \) and \( y = 1 \).

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Bonus Knowledge

To solve the system of equations, you can substitute the second equation into the first. Start with the equations: 1. \( -2x + 3y = 7 \) 2. \( y = 3x + 7 \) Now substitute \( y \) in the first equation: \[ -2x + 3(3x + 7) = 7 \] This simplifies to: \[ -2x + 9x + 21 = 7 \] \[ 7x + 21 = 7 \] Subtract 21 from both sides: \[ 7x = -14 \implies x = -2 \] Now substitute \( x \) back into the second equation to find \( y \): \[ y = 3(-2) + 7 = -6 + 7 = 1 \] Thus, the solution to the system of equations is \( x = -2 \) and \( y = 1 \). To verify, plug \( x \) and \( y \) back into the original equations: 1. \( -2(-2) + 3(1) = 4 + 3 = 7 \) (True) 2. \( 1 = 3(-2) + 7 = -6 + 7 = 1 \) (True) So the solution is confirmed!

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