Question

Determine whether the pair of lines represented by the equations are parallel, perpendicular, or neither. $$ 3 x-4 y=8 \text { and } 6 x-8 y=10 $$

Short answer

The lines represented by the equations \(3x - 4y = 8\) and \(6x - 8y = 10\) have the same slope (\(\frac{3}{4}\)), which means they are parallel.

Step-by-step solution

Find the slope of the first line

First, we need to find the slope of the first line. To do this, we will put the equation in the slope-intercept form, which is \(y = mx + b\), where m represents the slope and b is the y-intercept. The given equation for the first line is: \(3x - 4y = 8\) We will now solve for y. \(-4y = -3x + 8\) Now, divide by -4 to isolate y. \(y = \frac{3}{4}x - 2\) From this equation, we can see that the slope of the first line, \(m_1\), is \(\frac{3}{4}\).

Find the slope of the second line

Next, we need to find the slope of the second line. The given equation for the second line is: \(6x - 8y = 10\) We will follow the same steps as before to put it in slope-intercept form: \(-8y = -6x + 10\) Now, divide by -8 to isolate y. \(y = \frac{3}{4}x - \frac{5}{4}\) From this equation, we can see that the slope of the second line, \(m_2\), is also \(\frac{3}{4}\).

Determine the relationship between the lines

Now that we have the slopes of both lines, we can determine their relationship: 1. Since \(m_1 = m_2 = \frac{3}{4}\), the lines are parallel. 2. The product of the slopes is \(\frac{3}{4} \cdot \frac{3}{4}\), which is not equal to -1, so the lines are not perpendicular. Since the slopes are the same, the lines are parallel.