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below have a solution? Explain. \[ \begin{array}{l}\frac{3}{4} x-3=y \\ 4 x-3 y=5\end{array} \]

Ask by Chavez Griffin. in the United States
Feb 26,2025

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The solution to the system of equations is \( x = -\frac{16}{7} \) and \( y = -\frac{33}{7} \).

Solución

Solve the system of equations \( \frac{3}{4}x-3=y;4x-3y=5 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}\frac{3}{4}x-3=y\\4x-3y=5\end{array}\right.\) - step1: Rewrite the expression: \(\left\{ \begin{array}{l}y=\frac{3}{4}x-3\\4x-3y=5\end{array}\right.\) - step2: Substitute the value of \(y:\) \(4x-3\left(\frac{3}{4}x-3\right)=5\) - step3: Simplify: \(\frac{7}{4}x+9=5\) - step4: Move the constant to the right side: \(\frac{7}{4}x=5-9\) - step5: Subtract the numbers: \(\frac{7}{4}x=-4\) - step6: Multiply by the reciprocal: \(\frac{7}{4}x\times \frac{4}{7}=-4\times \frac{4}{7}\) - step7: Multiply: \(x=-\frac{16}{7}\) - step8: Substitute the value of \(x:\) \(y=\frac{3}{4}\left(-\frac{16}{7}\right)-3\) - step9: Calculate: \(y=-\frac{33}{7}\) - step10: Calculate: \(\left\{ \begin{array}{l}x=-\frac{16}{7}\\y=-\frac{33}{7}\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=-\frac{16}{7}\\y=-\frac{33}{7}\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(-\frac{16}{7},-\frac{33}{7}\right)\) The solution to the system of equations is \( (x,y) = (-\frac{16}{7},-\frac{33}{7}) \) or \( (x,y) = (-2.285714, -4.714285) \).

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Mind Expander

To determine if the equations have a solution, we can solve them simultaneously. The first equation can be rearranged to express \( y \) in terms of \( x \): \[ y = \frac{3}{4} x - 3. \] Next, we can substitute this expression for \( y \) into the second equation: \[ 4x - 3\left(\frac{3}{4}x - 3\right) = 5. \] Solving this gives us a value for \( x \), which we can then use to find \( y \). If there’s a consistent \( (x, y) \) solution, then the system has a solution. In this case, yes, the system has a unique solution! To find the solution, conduct the substitution and simplify the second equation, leading to a value of \( x \). Then plug it back to find \( y \).

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