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
$3x + 4y = 7$
$3x + 4y = 8$
$circ$ the system has no solution.
$circ$ the system has a unique solution:
$(x, y) = (\square, \square)$
$circ$ the system has infinitely many solutions.

system b
$y = 7x$
$y = x$
$circ$ the system has no solution.
$circ$ the system has a unique solution:
$(x, y) = (\square, \square)$
$circ$ the system has infinitely many solutions.

Explanation:

Step1: Analyze System A

We have the equations:
$3x + 4y = 7$
$3x + 4y = 8$
Subtract the first equation from the second:
$$(3x + 4y) - (3x + 4y) = 8 - 7$$
$$0 = 1$$
This is a false statement, meaning no solution exists.

Step2: Analyze System B

We have the equations:
$y = 7x$
$y = x$
Substitute $y = x$ into $y = 7x$:
$$x = 7x$$
Subtract $x$ from both sides:
$$0 = 6x$$
Solve for $x$:
$$x = 0$$
Substitute $x=0$ into $y=x$:
$$y = 0$$

Answer:

System A: The system has no solution.
System B: The system has a unique solution:
$(x, y) = (0, 0)$