Question

The lines $y=2x-1$and $y=ax+2$are perpendicular if$a=$____.

Short answer

The lines $y=2x-1$and $y=ax+2$are perpendicular if$a=-\frac{1}{2}$.

Step-by-step solution

Step 1. Given information.

We have two equation:

$$\begin{gathered} y=2x-1 \\ y=ax+2 \end{gathered}$$

Step 2. Fill the blank. 

Two non-vertical lines are perpendicular if and only if the product of their slopes is $-1.$

Comparing the two equation with slope intercept form $y=mx+b$, we get:

$$\begin{gathered} m_1=2 \\ {m}_2=a \end{gathered}$$

Since two lines are perpendicular, we get:

$$\begin{gathered} m_1\times m_2=-1 \\ 2\times a=-1 \\ a=-\frac{1}{2} \end{gathered}$$