Question

1. given the system of equations, determine whether the lines are parallel, perpendicular, or intersecting.$y + 3x = 7$$y = -3x -10$a the slopes are equivalent and therefore, there are no points at which these lines intersect, yielding no solutions for the system and therefore the lines are parallel.b the slopes are not equivalent and therefore, there exists a point at which these lines intersect, yielding at least one solution for the system.c the slopes are opposite reciprocals and therefore, they intersect at exactly one point and are perpendicular, yielding only one solution for the system.d the slopes are equivalent and therefore, there are no points at which these lines intersect, yielding no solutions for the system and therefore the lines are perpendicular.

Explanation:

Step1: Rewrite first equation to slope-intercept form

Rearrange $y + 3x = 7$ to $y = -3x + 7$.

Step2: Compare slopes of both lines

First line slope: $m_1 = -3$; Second line slope: $m_2 = -3$.

Step3: Compare y-intercepts

First line y-intercept: $b_1 = 7$; Second line y-intercept: $b_2 = -10$.

Step4: Classify the line relationship

Equal slopes, different y-intercepts mean parallel lines (no intersection, no solutions).

Answer:

A. The slopes are equivalent and therefore, there are no points at which these lines intersect, yielding no solutions for the system and therefore the lines are parallel.