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: \\ $y=$

Solution

Solution Steps

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

$-x + 4y = -8$
$x = 8 + 4y$

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:

$-x + 4y = -8$
$x = 4y + 8$

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.