Examine the system of equations.

Question

y = 4x + 8,

y = 4x – 1

Which statements best describe the two expressions? Check all that apply.
They have different slopes.
They have different y-intercepts.
The substitution method results in the false statement, 8 = –1.
The solution is (8, –1).
There is no solution.

Answer 1

To examine the system of equations:

1. Equations:
   • \( y = 4x + 8 \)
   • \( y = 4x - 1 \)

Let's analyze the statements provided:

1. They have different slopes:

   • Both equations have the same slope of 4 (the coefficient of \( x \)). Therefore, this statement is false.

2. They have different y-intercepts:

   • The first equation has a y-intercept of 8, while the second has a y-intercept of -1. Therefore, this statement is true.

3. The substitution method results in the false statement, 8 = –1:

   • Using substitution, we set the two equations equal to each other: \[ 4x + 8 = 4x - 1 \] Subtracting \( 4x \) from both sides gives: \[ 8 = -1 \] This is a false statement. Therefore, this statement is true.

4. The solution is (8, –1):

   • Since the two lines are parallel (same slope but different y-intercepts), they never intersect. Therefore, there is no solution, making this statement false.

5. There is no solution:

   • Since the lines are parallel and do not intersect, there is indeed no solution. Therefore, this statement is true.

Summary of the true statements:

• They have different y-intercepts.
• The substitution method results in the false statement, 8 = –1.
• There is no solution.

Final Answer:

• They have different y-intercepts.
• The substitution method results in the false statement, 8 = –1.
• There is no solution.