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Solve for \( x \) and \( y \) \( 30 x-42 y=-6 \) \( 5 x-7 y=-1 \)

Ask by Craig Love. in the United States
Jan 24,2025

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There are infinitely many solutions for \(x\) and \(y\) in the system of equations.

Solución

Solve the system of equations \( 30x-42y=-6;5x-7y=-1 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}30x-42y=-6\\5x-7y=-1\end{array}\right.\) - step1: Rearrange the terms: \(30x-42y=-6\) - step2: Move the expression to the right side: \(30x=-6+42y\) - step3: Divide both sides: \(\frac{30x}{30}=\frac{-6+42y}{30}\) - step4: Divide the numbers: \(x=\frac{-1+7y}{5}\) - step5: Calculate: \(\left(x,y\right) = \left(\frac{-1+7y}{5},y\right),y \in \mathbb{R}\) - step6: Alternative Form: \(\textrm{Infinitely many solutions}\) The system of equations \(30x-42y=-6\) and \(5x-7y=-1\) has infinitely many solutions. This means that there are an infinite number of pairs of values for \(x\) and \(y\) that satisfy both equations simultaneously.

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Mind Expander

To solve the system of equations 1. \( 30x - 42y = -6 \) 2. \( 5x - 7y = -1 \) we can manipulate the second equation by multiplying it by 6 to match the coefficients of \( x \) in the first equation: \( 6(5x - 7y) = 6(-1) \) Which gives us: \( 30x - 42y = -6 \) Since both equations are identical, we can see that they represent the same line, meaning there are infinitely many solutions along the line defined by these equations. To express \( y \) in terms of \( x \), we can manipulate the second equation: Starting from \( 5x - 7y = -1 \): 1. Isolate \( 7y \): \( 7y = 5x + 1 \) 2. Divide by 7: \( y = \frac{5}{7}x + \frac{1}{7} \) Thus, \( y \) can be expressed in terms of \( x \), providing a continuous set of solutions. You can substitute any value for \( x \) to find corresponding values of \( y \).

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