Example 1

What is the slope-intercept form of the equation of the line that is parallel to 3x + 2y = 8, and passes through (–2, 6)?

• Step 1
• Step 2

Rewrite 3x + 2y = 8 in slope-intercept form in order to identify the slope.

3x + 2y = 8   Subtract 3x from both sides of the equation.

2y = –3x + 8  Divide both sides of the equation by 2.

y = x + 4

The slope is .

Since parallel lines have the same slope, you are looking for the equation of a line with a slope of that passes through (–2, 6).

Substitute the slope and the point (–2, 6) into point-slope form and rewrite the equation in slope-intercept form.

y – y1 = m(x – x1)

y – 6 = [x – (–2)] Substitute the slope for m and the coordinates of the point for (x1, y1).

y – 6 = (x + 2)       Recall that the multiplication of two negatives simplifies to a positive.

y – 6 = x – 3         Distribute .

y = + 3                Add 6 to both sides of the equation.

The slope-intercept form of the equation of the line that is parallel to

3x + 2y = 8, and passes through (-2, 6), is y = + 3.

You can check this with a graph to verify that the lines are parallel.

Notice that both lines have the same slope, but have different y-intercepts.

Example 2

What is the slope-intercept form of the equation of the line that is perpendicular to y = x – 5 and passes through (3, 1)? Select each step to see the solution

• Step 1
• Step 2

Determine the slope of the line perpendicular to y = x – 5. Since the slope of the given line is the opposite reciprocal of that is −2. Because perpendicular lines have slopes that are opposite reciprocals, you are looking for the equation of a line with a slope of −2 that passes through (3, 1).

Substitute the slope and the point (3, 1) into point-slope form and rewrite the equation in slope-intercept form.

y − y1 = m(x – x1)

y − 1 = –2( x – 3)   Substitute the slope for m and the coordinates of the point for (x1, y1)

y − 1 = −2x + 6      Distribute –2.

y = −2x + 7           Add 1 to both sides of the equation.

The slope-intercept form of the equation of the line that is perpendicular to y = x – 5 and passes through (3, 1), is y = −2x + 7.

You can check this with a graph to verify that the lines are perpendicular.

Notice that the slopes of the lines are opposite reciprocals and cross at a 90° angle.