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+3 y =6 x-3 y =-6 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=

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+3 y =6
x-3 y =-6
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: 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 \[ \begin{aligned} -x+3 y & =6 \\ x-3 y & =-6 \end{aligned} \] 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: \[ y= \] $\square$

Solution

Solution Steps

Solution Approach

For each system of equations, we will use the method of elimination to determine the type of solution. By adding or subtracting the equations, we can eliminate one of the variables and solve for the other. If the resulting equation is true (like 0=0), the system has infinitely many solutions. If it results in a false statement (like 0=5), the system has no solution. Otherwise, we can find a unique solution for the variables.

Step 1: Analyze System A

For System A, we have the equations: \[ -x + 5y = 5 \quad (1) \] \[ x - 5y = 5 \quad (2) \] Upon solving these equations, we find that there is no solution, indicating that the lines represented by these equations are parallel and do not intersect.

Step 2: Analyze System B

For System B, we have the equations: \[ -x + 3y = 6 \quad (3) \] \[ x - 3y = -6 \quad (4) \] Solving these equations yields a relationship between $x$ and $y$: \[ x = 3y - 6 \] This indicates that there are infinitely many solutions, as the equations represent the same line.

Final Answer

For System A, the answer is that the system has no solution. For System B, the system has infinitely many solutions described by the equation $x = 3y - 6$.

Thus, the final answers are: