Question

1. natalie graphed a system of equations with no solution. what must be true about the equations of the lines?

a. they have the same slope.
b. they have different y-intercepts.
c. the lines are parallel.
d. all of the above.

Explanation:

Step1: Recall system of equations

A system of linear equations with no solution means the lines never intersect.

Step2: Analyze slope and intercept

• For two lines \( y = m_1x + b_1 \) and \( y = m_2x + b_2 \), if they are parallel, \( m_1=m_2 \) (same slope).
• If they had the same \( y \)-intercept and same slope, they would be the same line (infinitely many solutions). So for no solution, they must have different \( y \)-intercepts (so they are parallel, not coinciding) and same slope.
• Option a: Same slope is required for parallel (no intersection).
• Option b: Different \( y \)-intercepts (otherwise, same line with infinite solutions).
• Option c: Lines are parallel (which requires same slope and different \( y \)-intercepts).
• So all (a, b, c) are true, hence d.

Answer:

d. All of the above.