Questions: System A -x+4 y = -8 x = 8+4 y The system has no solution. The system has a unique solution: (x, y)= The system has infinitely many solutions. They must satisfy the following equation: y=

System A -x+4 y = -8 x = 8+4 y The system has no solution. The system has a unique solution: (x, y)= The system has infinitely many solutions. They must satisfy the following equation: y=
Transcript text: System $A$ \\ \begin{aligned}-x+4 y & =-8 \\ x & =8+4 y\end{aligned} \\ The system has no solution. \\ The system has a unique solution: \\ $(x, y)=(\square$ \\ ) \\ The system has infinitely many solutions. \\ They must satisfy the following equation: \\ $y=$
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Solution

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Solution Steps

To determine the nature of the solutions for the given system of equations, we need to analyze the equations. The system is:

  1. \(-x + 4y = -8\)
  2. \(x = 8 + 4y\)

First, we can substitute the expression for \(x\) from the second equation into the first equation to see if there is a contradiction, a unique solution, or if the equations are dependent (leading to infinitely many solutions).

Step 1: Analyze the System of Equations

We have the system of equations:

  1. \(-x + 4y = -8\)
  2. \(x = 4y + 8\)
Step 2: Substitute and Simplify

Substitute the expression for \(x\) from the second equation into the first equation: \[ -x + 4y = -8 \quad \Rightarrow \quad -(4y + 8) + 4y = -8 \]

Simplify the equation: \[ -4y - 8 + 4y = -8 \]

Step 3: Determine the Nature of the System

The simplified equation becomes: \[ -8 = -8 \]

This is a true statement, indicating that the original equations are dependent and represent the same line. Therefore, the system has infinitely many solutions.

Final Answer

The system has infinitely many solutions. They must satisfy the equation: \[ x = 4y + 8 \] \(\boxed{x = 4y + 8}\)

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