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5. \( \left\{\begin{array}{l}-3 x+3 y=-3 \\ x-3 y=-1\end{array}\right. \)

Ask by Long Boyd. in the United States
Mar 14,2025

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Tutor-Verified Answer

Answer

The solution is \( x=2 \) and \( y=1 \).

Solution

Solve the system of equations \( -3x+3y=-3;x-3y=-1 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}-3x+3y=-3\\x-3y=-1\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}-3x+3y=-3\\x=-1+3y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(-3\left(-1+3y\right)+3y=-3\) - step3: Simplify: \(3-6y=-3\) - step4: Move the constant to the right side: \(-6y=-3-3\) - step5: Subtract the numbers: \(-6y=-6\) - step6: Change the signs: \(6y=6\) - step7: Divide both sides: \(\frac{6y}{6}=\frac{6}{6}\) - step8: Divide the numbers: \(y=1\) - step9: Substitute the value of \(y:\) \(x=-1+3\times 1\) - step10: Substitute back: \(x=-1+3\) - step11: Calculate: \(x=2\) - step12: Calculate: \(\left\{ \begin{array}{l}x=2\\y=1\end{array}\right.\) - step13: Check the solution: \(\left\{ \begin{array}{l}x=2\\y=1\end{array}\right.\) - step14: Rewrite: \(\left(x,y\right) = \left(2,1\right)\) The solution to the system of equations is \( (x,y) = (2,1) \).

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The Deep Dive

To solve the system of equations, let's rewrite them: 1. \(-3x + 3y = -3\) simplifies to \(y = x - 1\) 2. \(x - 3y = -1\) can be rearranged to \(x = 3y - 1\) Now, substitute \(y = x - 1\) into the second equation: \(x = 3(x - 1) - 1\) This becomes: \(x = 3x - 3 - 1\) which simplifies to: \(x = 3x - 4\) Rearranging gives: \(2x = 4\) Thus, \(x = 2\). Now, substitute \(x\) back into \(y = x - 1\): \(y = 2 - 1 = 1\) So, the solution to the system is \(x = 2\) and \(y = 1\). Therefore, the final answer is \( (2, 1) \). In a historical context, the method of solving systems of equations, like those we've just tackled, dates back to ancient civilizations. The Chinese, for example, used a method called "fangcheng" during the Han Dynasty, and the famous mathematician Liu Hui laid early groundwork for these types of calculations around 200 AD. When applying the methods to real-world scenarios, systems of equations often represent constraints or relationships. For instance, in finance, they can help determine the best mix of investments to meet a budget while maximizing returns, highlighting how math plays a vital role in decision-making in various fields!

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