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
y=5x+8
y=5x+3

• the system has no solution.
• the system has a unique solution:

(x,y)= (□,□)

• the system has infinitely many solutions.

system b
-x-3y=1
x+3y=-1

• the system has no solution.
• the system has a unique solution:

(x,y)= (□,□)

• the system has infinitely many solutions.

Explanation:

Step1: Analyze System A

The two equations in System A are \( y = 5x + 8 \) and \( y = 5x + 3 \). These are linear equations in slope - intercept form (\( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept). Both equations have the same slope (\( m = 5 \)) but different y - intercepts (\( b = 8 \) and \( b = 3 \)). Parallel lines (lines with the same slope and different y - intercepts) never intersect, so the system has no solution.

Step2: Analyze System B

The equations in System B are \( -x-3y = 1 \) and \( x + 3y=-1 \). Let's rewrite the first equation: \( -x-3y = 1\Rightarrow x + 3y=-1 \). So both equations are identical. When two equations in a system of linear equations are identical, the lines coincide (they are the same line), and there are infinitely many solutions (all the points on the line are solutions).

Answer:

System A:

The system has no solution.

System B:

The system has infinitely many solutions.