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9. \( \begin{array}{l}y=\frac{3}{4} x+5 \\ -3 x+4 y=2\end{array} \)

Ask by Parsons Wyatt. in the United States
Mar 14,2025

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The system of equations has no solution.

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Solve the system of equations \( y=\frac{3}{4}x+5;-3x+4y=2 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}y=\frac{3}{4}x+5\\-3x+4y=2\end{array}\right.\) - step1: Substitute the value of \(y:\) \(-3x+4\left(\frac{3}{4}x+5\right)=2\) - step2: Simplify: \(20=2\) - step3: Calculate: \(\left\{ \begin{array}{l}x \in \varnothing \\y \in \varnothing \end{array}\right.\) - step4: Rewrite: \((x, y) \in \varnothing\) The system of equations \( y=\frac{3}{4}x+5 \) and \( -3x+4y=2 \) has no solution.

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To solve this system of equations, let's first rearrange the second equation into slope-intercept form. Starting with \(-3x + 4y = 2\), we can isolate \(y\): 1. Add \(3x\) to both sides: \(4y = 3x + 2\) 2. Divide everything by 4: \(y = \frac{3}{4}x + \frac{1}{2}\) Now, we have two equations: 1. \(y = \frac{3}{4}x + 5\) 2. \(y = \frac{3}{4}x + \frac{1}{2}\) Next, observe that both equations have the same slope of \(\frac{3}{4}\). This means they are parallel lines. Since parallel lines never intersect, this system has no solution! In this case, the lines will never meet no matter how far you extend them. Isn’t it interesting how algebra can tell us about the relationships between lines graphically? If only all problems were so straightforward!

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