Click and drag the points on the red line. The points on the y-axis will only move up and down, and the other point can move anywhere.

The blue line can be changed by editing the slope and y-intercept in the boxes below.

- Move the red line until you get the message "a and b are parallel lines". What do you notice about the slopes when the two lines are parallel?
- Can you create two different parallel lines with the same y-intercept?
- What do you notice about the slopes when the two lines are perpendicular? Move the red line until you get the message "a and b are perpendicular lines".
- Can you create two distinct perpedicular lines with the same y-intercept?

Consider a line perpendicular to y = 3x-2 with the same y-intercept. We know the slope must be the negative reciprocal which is m = -\dfrac{1}{3} and the y-intercept must be the same, which is b = -2.

Therefore, the equation of a perpendicular line with the same y-intercept is y = -\dfrac{1}{3}x - 2.

The line AB passes through the points \left(-2,\,9\right) and \left(3,\,-21\right).

a

Write the equation of the line.

Worked Solution

b

Find the equation of the line that passes through \left(1,\,5\right) and is parallel to the line AB.

Worked Solution

Consider the line 4x-3y=-6.

a

Find the equation of the line that is perpendicular to the given line and has the same y-intercept.

Worked Solution

b

Write the equation in standard form.

Worked Solution

c

Determine the zeros of the linear function.

Worked Solution

A mirror is placed along the x-axis. A laser beam is projected along the line y=-x+4 which reflects off the mirror.

a

A *normal* is a line which is perpendicular to the surface of the mirror at the point of reflection. Find the equation of the normal.

Worked Solution

b

The angles that the laser and its reflection make with the normal will be congruent. If the angle between the laser beam and the normal is 45 \degree, find the equation of the path of the reflection.

Worked Solution

Idea summary

If two lines are parallel, they have the same slope.

If two lines have the same slope, they are parallel.

The slopes of perpendicular lines are reciprocals with opposite signs. When multiplied together, they have a product of -1.