Question

what do the following two equations represent?

• $y - 2 = x + 20$
• $y - 2 = -(x + 20)$

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

Explanation:

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope.

For the first equation $y - 2=x + 20$, we can rewrite it as $y - 2 = 1\times(x+20)$. Comparing with the point - slope form $y - y_1=m(x - x_1)$ (here $x_1=- 20,y_1 = 2$), the slope $m_1=1$.

Step2: Find the slope of the second equation

For the second equation $y - 2=-(x + 20)$, we can rewrite it as $y - 2=-1\times(x + 20)$. Comparing with the point - slope form $y - y_1=m(x - x_1)$ (here $x_1=-20,y_1 = 2$), the slope $m_2=- 1$.

Step3: Analyze the relationship between the slopes

We know that two lines are perpendicular if the product of their slopes $m_1\times m_2=-1$.

Calculate $m_1\times m_2=(1)\times(-1)=-1$. Also, both lines pass through the point $(-20,2)$ (since when $x=-20$, for the first equation: $y-2=-20 + 20=0\Rightarrow y = 2$; for the second equation: $y - 2=-(-20 + 20)=0\Rightarrow y = 2$). So the two lines intersect at the point $(-20,2)$ and their slopes multiply to - 1, which means they are perpendicular.

Answer:

C. Perpendicular lines