Question

graph the two equations in the same coordinate plane. Use the following information to decide whether the lines are perpendicular. It can be shown that two different nonvertical lines in the same plane with slopes \(m_{1}\) and \(m_{2}\) are perpendicular if and only if \(m_{2}\) is the negative reciprocal of \(m_{1}\) $$ \begin{aligned} &y=2 x+4\\\ &y=-\frac{1}{2} x+2 \end{aligned} $$

Short answer

Yes, the two lines are perpendicular. The slopes of the equations have been calculated as 2 and -1/2, satisfying the condition for lines to be perpendicular, which is that one slope must be the negative reciprocal of the other.

Step-by-step solution

Identification of Slopes

From the given equations, the slope of the first line, \(m_1\), can be identified as 2 (the coefficient of \(x\)) in the equation \(y=2x+4\). Similarly, for the second line, the slope, \(m_2\), is -1/2 in the equation \(y=-\frac{1}{2}x+2\).

Graphing the Equations

The equations would then be graphed on the same coordinate plane. This involves plotting the equations \(y=2x+4\) and \(y=-\frac{1}{2}x+2\).

Verification of Perpendicularity

With the slopes identified, the condition for perpendicularly is checked. Two lines are said to be perpendicular if and only if the slope of one line is the negative reciprocal of the other. In this case, as \(m_1=2\) and \(m_2=-\frac{1}{2}\), it can be observed that \(m_2\) is the negative reciprocal of \(m_1\). Therefore, the lines are perpendicular.