Questions: Two systems of equations are given below. For each system, choose the best description of its solution. If applicable, give the solution. System A -x+5 y=-5 x+5 y=-5 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 B x-4 y =-8 -x+4 y+8 =0 The system has no solution. The system has a unique solution: (x, y)= The system has infinitely many solutions.

Two systems of equations are given below.
For each system, choose the best description of its solution. If applicable, give the solution.

System A
-x+5 y=-5
x+5 y=-5
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 B
x-4 y =-8
-x+4 y+8 =0
The system has no solution.
The system has a unique solution:
(x, y)=

The system has infinitely many solutions.

Transcript text: Two systems of equations are given below. For each system, choose the best description of its solution. If applicable, give the solution. System A \[ \begin{array}{r} -x+5 y=-5 \\ x+5 y=-5 \end{array} \] The system has no solution. The system has a unique solution: \[ (x, y)=(\square, \square) \] The system has infinitely many solutions. They must satisfy the following equation: \[ y= \] $\square$ System B \[ \begin{aligned} x-4 y & =-8 \\ -x+4 y+8 & =0 \end{aligned} \] The system has no solution. The system has a unique solution: \[ (x, y)= \] $\square$ $\square$ The system has infinitely many solutions.

Solution

Solution Steps

To solve the given systems of equations, we will use the method of elimination or substitution to determine if the system has no solution, a unique solution, or infinitely many solutions.

System A

Add the two equations to eliminate $x$ and solve for $y$.
Substitute $y$ back into one of the original equations to solve for $x$.
Check if the solution is consistent or if there are contradictions.

System B

Simplify the second equation and compare it with the first equation.
Determine if the equations are consistent, contradictory, or dependent.

Step 1: Analyze System A

For System A, we have the equations: \[ -x + 5y = -5 \quad (1) \] \[ x + 5y = -5 \quad (2) \]

By adding equations (1) and (2), we eliminate $x$: \[ (-x + 5y) + (x + 5y) = -5 - 5 \] This simplifies to: \[ 10y = -10 \implies y = -1 \]

Substituting $y = -1$ back into equation (1): \[ -x + 5(-1) = -5 \implies -x - 5 = -5 \implies -x = 0 \implies x = 0 \]

Thus, the solution for System A is: \[ (x, y) = (0, -1) \]

Step 2: Analyze System B

For System B, we have the equations: \[ x - 4y = -8 \quad (3) \] \[ -x + 4y + 8 = 0 \quad (4) \]

Rearranging equation (4): \[ -x + 4y = -8 \implies x - 4y = 8 \]

Now we have: \[ x - 4y = -8 \quad (3) \] \[ x - 4y = 8 \quad (5) \]

Since the left-hand sides of equations (3) and (5) are identical but the right-hand sides are different, this indicates a contradiction. Therefore, System B has no solution.