Question

question 4
1 pts
in the system y=3x+5 and y=3x-7, the lines are:
○ coinciding
○ intersecting
○ parallel
○ perpendicular

Explanation:

Step1: Recall slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.
For the line $y = 3x+5$, the slope $m_1=3$ and the y - intercept $b_1 = 5$.
For the line $y=3x - 7$, the slope $m_2 = 3$ and the y - intercept $b_2=-7$.

Step2: Analyze the slopes and y - intercepts

• Coinciding lines: Coinciding lines have the same slope and the same y - intercept. Here, $b_1 = 5$ and $b_2=-7$, so they are not coinciding.
• Intersecting lines: Intersecting lines have different slopes (or if slopes are equal, they must be coinciding, but here slopes are equal and y - intercepts are different). For two lines $y = m_1x + b_1$ and $y=m_2x + b_2$, if $m_1

eq m_2$, they intersect. If $m_1 = m_2$ and $b_1
eq b_2$, they are parallel, not intersecting.

• Parallel lines: Two lines in the slope - intercept form are parallel if their slopes are equal ($m_1=m_2$) and their y - intercepts are different ($b_1

eq b_2$). Since $m_1 = m_2=3$ and $b_1 = 5
eq b_2=-7$, the lines are parallel.

• Perpendicular lines: Two lines are perpendicular if the product of their slopes is $- 1$ (i.e., $m_1\times m_2=-1$). Here, $m_1\times m_2=3\times3 = 9

eq - 1$, so they are not perpendicular.

Answer:

parallel (the option corresponding to "parallel")