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What is the solution to the system of equations? \( y=\frac{1}{8} x-1 \) \( -5 x+4 y=-13 \)

Ask by Frazier Murphy. in the United States
Jan 22,2025

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The solution to the system of equations is \( x = 2 \) and \( y = -0.75 \).

Solución

Solve the system of equations \( y=\frac{1}{8}x-1;-5x+4y=-13 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}y=\frac{1}{8}x-1\\-5x+4y=-13\end{array}\right.\) - step1: Substitute the value of \(y:\) \(-5x+4\left(\frac{1}{8}x-1\right)=-13\) - step2: Simplify: \(-\frac{9}{2}x-4=-13\) - step3: Move the constant to the right side: \(-\frac{9}{2}x=-13+4\) - step4: Add the numbers: \(-\frac{9}{2}x=-9\) - step5: Change the signs: \(\frac{9}{2}x=9\) - step6: Multiply by the reciprocal: \(\frac{9}{2}x\times \frac{2}{9}=9\times \frac{2}{9}\) - step7: Multiply: \(x=2\) - step8: Substitute the value of \(x:\) \(y=\frac{1}{8}\times 2-1\) - step9: Simplify: \(y=-\frac{3}{4}\) - step10: Calculate: \(\left\{ \begin{array}{l}x=2\\y=-\frac{3}{4}\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=2\\y=-\frac{3}{4}\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(2,-\frac{3}{4}\right)\) The solution to the system of equations is \( (x,y) = (2,-\frac{3}{4}) \) or \( (x,y) = (2,-0.75) \).

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The Deep Dive

To solve the system of equations, start by substituting the expression for \( y \) from the first equation into the second equation. This gives us: \[ -5x + 4\left(\frac{1}{8}x - 1\right) = -13 \] Next, distribute the 4: \[ -5x + \frac{4}{8}x - 4 = -13 \] which simplifies to: \[ -5x + \frac{1}{2}x - 4 = -13 \] Now, combine the \( x \) terms: \[ -\frac{10}{2}x + \frac{1}{2}x = -13 + 4 \] This simplifies to: \[ -\frac{9}{2}x = -9 \] Now, to isolate \( x \), multiply both sides by \(-\frac{2}{9}\): \[ x = 2 \] Now that you have \( x \), substitute it back into the first equation to find \( y \): \[ y = \frac{1}{8}(2) - 1 = \frac{1}{4} - 1 = -\frac{3}{4} \] So the solution to the system of equations is: \[ (x, y) = (2, -\frac{3}{4}) \]

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