Question

decide if the two lines y = 8x-6 and y=8x +8 are parallel perpendicular or neither

Answer 1

1) Two lines are parallel when they have the same slope, for example, the lines:

$\begin{gathered} y_1=2x_1+3 \\ y_2=2x_2-5 \end{gathered}$

The coefficients of the x-terms of the lines are their slopes. In this example, both slopes are equal to 2, so the lines are parallel.

$m_1=m_2=2$

m₁ indicates the slope of the first line

m₂ indicates the slope of the second line

2) If two lines are perpendicular, then their slopes are reverse opposites, for example, given the lines:

$\begin{gathered} y_1=m_1x_1+b_1 \\ y_2=m_2x_2+b_2 \end{gathered}$

For both lines to be perpendicular the relationship between their slopes must be the following:

$m_2=-\frac{1}{m_1}$

If a line has a slope m₁=3, then the slope of the perpendicular line will be:

$\begin{gathered} m_2=-\frac{1}{m_1}_{} \\ m_2=-\frac{1}{3} \end{gathered}$

3) If the slopes are not equal nor reverse opposites, then the lines you are comparing are neither parallel nor perpendicular, for example, the lines:

$\begin{gathered} y_1=5x_1+9_{} \\ y_2=-\frac{1}{7}x-5 \end{gathered}$

With this in mind, considering the given lines:

$\begin{gathered} y=8x-6 \\ y=8x+8 \end{gathered}$

The slope of both lines is equal to 8, which means that the lines are parallel.