Question
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4. \( \left\{\begin{array}{l}6 y=6 x+24 \\ -3 x+3 y=12\end{array}\right. \)

Ask by Malone Barber. in the United States
Mar 14,2025

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Tutor-Verified Answer

Answer

The system has infinitely many solutions.

Solution

Solve the system of equations \( 6y=6x+24;-3x+3y=12 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}6y=6x+24\\-3x+3y=12\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=y-4\\-3x+3y=12\end{array}\right.\) - step2: Substitute the value of \(x:\) \(-3\left(y-4\right)+3y=12\) - step3: Simplify: \(12=12\) - step4: Calculate: \(\left(x,y\right) = \left(y-4,y\right),y \in \mathbb{R}\) - step5: Alternative Form: \(\textrm{Infinitely many solutions}\) The system of equations has infinitely many solutions. This means that the two equations are linearly dependent, and there are an infinite number of solutions that satisfy both equations.

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Simplify this solution

The Deep Dive

Let’s have some fun solving this system of equations! First, we can simplify the equations a bit. The first equation \(6y = 6x + 24\) can be simplified by dividing everything by 6, giving us \(y = x + 4\). Now, substitute \(y\) into the second equation \(-3x + 3y = 12\). Replacing \(y\), we have: \[ -3x + 3(x + 4) = 12. \] Now, combine like terms: \[ -3x + 3x + 12 = 12. \] This simplifies to \(12 = 12\), which means the equations represent the same line, indicating an infinite number of solutions! So any point of the form \((x, x + 4)\) is a solution to this system. For those learning to solve linear equations, remembering that two equations could represent the same line is a key concept!

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