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Graphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither · Physics Playground

Exercises Equations and inequalities

Example 14 from Algebra and Trigonometry, 2.2 Linear Equations in One Variable

Graphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither

Graph the equations of the given lines, and state whether they are parallel, perpendicular, or neither: $3 y = - 4 x + 3$ and $3 x - 4 y = 8.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The first thing we want to do is rewrite the equations so that both equations are in slope-intercept form.

First equation:

$\begin{matrix} 3 y & = & −4 x + 3 \\ y & = & - \frac{4}{3} x + 1 \end{matrix}$

Second equation:

$\begin{matrix} 3 x - 4 y & = & 8 \\ −4 y & = & -3x+8 \\ y & = & \frac{3}{4} x - 2 \end{matrix}$
See the graph of both lines in Figure 6

Figure 6

From the graph, we can see that the lines appear perpendicular, but we must compare the slopes.

$\begin{matrix} m_{1} & = & - \frac{4}{3} \\ m_{2} & = & \frac{3}{4} \\ m_{1} \cdot m_{2} & = & (- \frac{4}{3}) (\frac{3}{4}) = −1 \end{matrix}$
The slopes are negative reciprocals of each other, confirming that the lines are perpendicular.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Graph the two lines and determine whether they are parallel, perpendicular, or neither: $2 y - x = 10$ and $2 y = x + 4.$

How did it go?

PreviousFinding the Equation of a Line Passing Through the Given Points Next Writing the Equation of a Line Parallel to a Given Line Passing Through a Given Point

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 2.2, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.

Solution

Your turn