Question
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Find the solution to the system of equations. You can use the interactive graph below to find the solution. \( \left\{\begin{array}{l}-2 x+2 y=-4 \\ 3 x+3 y=-18\end{array}\right. \) \( x=\square \)

Ask by Cervantes Bradley. in the United States
Jan 10,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

\( x = -2 \)

Solution

Solve the system of equations \( -2x+2y=-4;3x+3y=-18 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}-2x+2y=-4\\3x+3y=-18\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=2+y\\3x+3y=-18\end{array}\right.\) - step2: Substitute the value of \(x:\) \(3\left(2+y\right)+3y=-18\) - step3: Simplify: \(6+6y=-18\) - step4: Move the constant to the right side: \(6y=-18-6\) - step5: Subtract the numbers: \(6y=-24\) - step6: Divide both sides: \(\frac{6y}{6}=\frac{-24}{6}\) - step7: Divide the numbers: \(y=-4\) - step8: Substitute the value of \(y:\) \(x=2-4\) - step9: Simplify: \(x=-2\) - step10: Calculate: \(\left\{ \begin{array}{l}x=-2\\y=-4\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=-2\\y=-4\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(-2,-4\right)\) The solution to the system of equations is \( x = -2 \) and \( y = -4 \).

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The Deep Dive

First, let's simplify the given equations. For the first equation, \(-2x + 2y = -4\), we can divide everything by 2 to get \(y = x - 2\). Now for the second equation, \(3x + 3y = -18\), we can divide by 3, resulting in \(x + y = -6\). Substituting \(y\) from the first simplified equation into the second gives us \(x + (x - 2) = -6\). Solving this, you’ll find that \(2x - 2 = -6\) leads to \(2x = -4\), thus \(x = -2\). Now, looking back to find \(y\), substituting \(-2\) back into \(y = x - 2\) gives \(y = -2 - 2 = -4\). Therefore, the solution to the system of equations is \(x = -2\) and \(y = -4\). To sum it up clearly: \(x = -2\) \(y = -4\)

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