Questions: The system has a unique solution: System B 2x+y=8 -2x-y+8=0 (x, y)=(, )

Transcript text: The system has a unique solution: System B $2 x+y=8$ $-2 x-y+8=0$ $(x, y)=(\square, \square)$

Solution

Solution Steps

To determine the nature of the solutions for the given system of linear equations, we need to analyze the equations and solve for $x$ and $y$. We can use methods such as substitution or elimination to find the solutions.

Solution Approach

Write down the given system of equations.
Use the elimination method to eliminate one of the variables.
Solve for the remaining variable.
Substitute back to find the other variable.
Check if the system has a unique solution, no solution, or infinitely many solutions.

Step 1: Analyze the System of Equations

We are given the system of equations: \[ \begin{cases} 2x + y = 8 \\ -2x - y + 8 = 0 \end{cases} \]

Step 2: Simplify the Second Equation

First, let's simplify the second equation: \[ -2x - y + 8 = 0 \] Subtract 8 from both sides: \[ -2x - y = -8 \]

Step 3: Add the Equations

Now, add the two equations together to eliminate $ y $: \[ (2x + y) + (-2x - y) = 8 + (-8) \] This simplifies to: \[ 0 = 0 \]

Step 4: Interpret the Result

The result $ 0 = 0 $ indicates that the two equations are dependent, meaning they represent the same line. Therefore, the system has infinitely many solutions.

Step 5: Express $ y $ in Terms of $ x $

To find the equation that represents the solutions, we can solve one of the original equations for $ y $. Let's use the first equation: \[ 2x + y = 8 \] Solve for $ y $: \[ y = 8 - 2x \]