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Chapter 1: Problem 67

Determine whether the lines are parallel, perpendicular, or neither. $$\begin{aligned} &L_{1}: y=\frac{1}{2} x-3\\\ &L_{2}: y=-\frac{1}{2} x+1 \end{aligned}$$

Short Answer

Expert verified
The lines \(L_1\) and \(L_2\) are perpendicular.

Step by step solution

01

Identify the Slope of Line \(L_1\)

In the equation \(y=\frac{1}{2}x - 3\), the slope of \(L_1\) is the coefficient of \(x\), that is \(\frac{1}{2}\).
02

Identify the Slope of Line \(L_2\)

The equation for line \(L_2\) is \(y= -\frac{1}{2}x + 1\). Just like in step 1, we identify that the coefficient of \(x\), here -\(\frac{1}{2}\), is the slope of \(L_2\).
03

Compare the Slopes

It is observed that the slope of \(L_2\) is the negative reciprocal of the slope of \(L_1\), as \(-\frac{1}{2}\) is the negative reciprocal of \(\frac{1}{2}\). According to property of lines, two lines are said to be perpendicular if the product of their slopes is -1. In this case, the product of \(\frac{1}{2}\) and -\(\frac{1}{2}\) is -1. Therefore, the lines are perpendicular.

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