Question

college algebra > unit 2
unit test
and forms of linear equations
what do the following two equations represent?

• $y - 8 = \frac{1}{2}(x + 5)$
• $y - 8 = -\frac{1}{2}(x + 5)$

choose 1 answer:
a the same line
b distinct parallel lines
c perpendicular lines
d intersecting, but not perpendicular lines

Explanation:

Step1: Identify slopes of lines

The equations are in point-slope form $y-y_1=m(x-x_1)$, where $m$ is the slope.
For $y-8=\frac{1}{2}(x+5)$, slope $m_1=\frac{1}{2}$.
For $y-8=-\frac{1}{2}(x+5)$, slope $m_2=-\frac{1}{2}$.

Step2: Check slope relationships

Parallel lines have equal slopes; perpendicular lines have slopes that multiply to $-1$.
$m_1 \times m_2 = \frac{1}{2} \times (-\frac{1}{2}) = -\frac{1}{4}
eq -1$, so not perpendicular.
$m_1
eq m_2$, so not parallel.

Step3: Verify intersection

Since slopes are different, the lines intersect at exactly one point, and they are not perpendicular.

Answer:

D. Intersecting, but not perpendicular lines