Question

two systems of equations are given below. for each system, choose the best description of its solution. if applicable, give the solution.
system a
$x + 5y = 9$
$-x + 5y = 1$
\bigcirc the system has no solution.
\bigcirc the system has a unique solution:
$(x, y) = (\square, \square)$
\bigcirc the system has infinitely many solutions.
system b
$-x - 4y = 2$
$x + 4y = -2$
\bigcirc the system has no solution.
\bigcirc the system has a unique solution:
$(x, y) = (\square, \square)$
\bigcirc the system has infinitely many solutions.

Explanation:

Step1: Solve System A

We have the equations:

1. \( x + 5y = 9 \)
2. \( -x + 5y = 1 \)

Add the two equations together to eliminate \( x \):
\( (x + 5y) + (-x + 5y) = 9 + 1 \)
\( 10y = 10 \)
Divide both sides by 10: \( y = 1 \)

Substitute \( y = 1 \) into the first equation \( x + 5(1) = 9 \)
\( x + 5 = 9 \)
Subtract 5 from both sides: \( x = 4 \)
So System A has a unique solution \( (x, y) = (4, 1) \)

Step2: Solve System B

We have the equations:

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

Add the two equations together:
\( (-x - 4y) + (x + 4y) = 2 + (-2) \)
\( 0 = 0 \)
This means the two equations are equivalent (they represent the same line), so there are infinitely many solutions.

Answer:

System A:

The system has a unique solution: \( (x, y) = (4, 1) \)

System B:

The system has infinitely many solutions.