Question

1. what does it mean if two lines in a system of equations are parallel?

a. they intersect at multiple points.
b. they have no points of intersection.
c. they are the same line.
d. they intersect at one point.

1. if the system of equations ( y = 3x + 5 ) and ( y = 3x + 5 ) is solved, what type of solution do we expect

a. no solution
b. one solution
c. undefined solution
d. infinitely many solutions

Explanation:

10.

Step1: Recall the property of parallel lines

Parallel lines never meet. In a system of equations (linear equations represent lines), if two lines are parallel, they have no common point.

Step2: Analyze each option

• Option a: Parallel lines do not intersect at multiple points.
• Option b: Since parallel lines never meet, they have no points of intersection.
• Option c: If they were the same line, they would be coincident, not parallel.
• Option d: Lines that intersect at one point are not parallel.

11.

Step1: Analyze the given equations

The equations are \(y = 3x+5\) and \(y = 3x + 5\). They are the same equation (represent the same line).

Step2: Recall the solution concept for systems of equations

When two equations represent the same line, every point on that line is a solution. So there are infinitely many solutions.

• Option a: No solution occurs when lines are parallel (different \(y\)-intercepts but same slope). Here, the equations are same.
• Option b: One solution occurs when lines intersect (different slopes).
• Option c: “Undefined solution” is not a standard term for the number of solutions of a system of linear equations.
• Option d: Since the equations are identical, there are infinitely many solutions.

Answer:

1. B. They have no points of intersection.
2. D. Infinitely many solutions.