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17Calculus Precalculus - Equations of Lines

Algebra

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Applications

Tools

Articles

Line Equations

slope

\(\displaystyle{ m = \frac{y_2 - y_1}{x_2 - x_1} }\)

point-slope form

\(y-y_1 = m(x-x_1)\)

slope-intercept form

\(y=mx+b\)

general form

\( Ax + By = C \)

Before we get down to the details of lines, let's watch this video to remind us of the slope of a line.

Prof Leonard - Introduction to the Slope of a Line [12min-29secs]

video by Prof Leonard

In order to determine an equation of a line, you need only two pieces of information, the slope and one point on the line. If you are given two points and no slope, use the slope equation \(\displaystyle{ m = \frac{y_2 - y_1}{x_2 - x_1} }\) to find the slope and then use one of the points (either one, its doesn't matter, you will get the same answer no matter which point you choose).

Once you have the slope, usually called m, and one point, \((x_1,y_1)\), you can find the equation of the line using two possible equations.
1. The one that is the easiest and requires the least amount of algebra is the point-slope form, \(y-y_1 = m(x-x_1)\). Notice that this is just the slope equation in a little different form.

Prof Leonard - Using the Point-Slope Equation of a Line [24min-6secs]

video by Prof Leonard

2. The second, which works too, is to use the slope-intercept form, \(y=mx+b\) and solve for \(b\). This seems to be the preferred method of most students.
For your final answer, most instructors prefer the slope-intercept form \(y=mx+b\). This is the required form for answers on this site. However, you need to check with your instructor to see what they expect. Here is a video on why we need the slope-intercept form.

Prof Leonard - Why We Need Slope-Intercept Form [23min-55secs]

video by Prof Leonard

Slope-Intercept Form of a Line

Adjust the sliders for the slope m and the y-intercept b to see the plot and the equation change.

As we said above, your goal is to get the slope and one point on the line in order to find the equation of the line. Start with these problems to get some practice on this.

Instructions - - Unless otherwise instructed, find the equation of the line, in slope-intercept form, with the given slope passing through the given point.

Find the equation of the line, in slope-intercept form, with slope \( -2/3 \) passing through the point \( (-4, 6) \).

Problem Statement

Find the equation of the line, in slope-intercept form, with slope \( -2/3 \) passing through the point \( (-4, 6) \).

Final Answer

\( y = -2x/3 +10/3 \)

Problem Statement

Find the equation of the line, in slope-intercept form, with slope \( -2/3 \) passing through the point \( (-4, 6) \).

Solution

2710 video

Final Answer

\( y = -2x/3 +10/3 \)

close solution

Find the equation of the line, in slope-intercept form, with slope \( -2/3 \) passing through the point \( (1, -1) \).

Problem Statement

Find the equation of the line, in slope-intercept form, with slope \( -2/3 \) passing through the point \( (1, -1) \).

Final Answer

\( y = -2x/3 - 1/3 \)

Problem Statement

Find the equation of the line, in slope-intercept form, with slope \( -2/3 \) passing through the point \( (1, -1) \).

Solution

We are given the slope \( m = -2/3 \) and a point on the line \( (1, -1) \).

Let's use the point-slope form of the equation of the line, \( y-y_1 = m(x-x_1) \)

\( y-(-1) = (-2/3)( x- 1) \)

\( y + 1 = -2x/3 + 2/3 \)

\( y = -2x/3 + 2/3 - 1 \)

Final Answer: \( y = -2x/3 - 1/3 \)

Here is how we would use the slope-intercept form to solve this problem.

\( y = mx + b \to -1 = (-2/3)(1) + b\)

\( b = -1 + 2/3 = -1/3 \)

Final Answer: \( y = -2x/3 - 1/3 \)

Final Answer

\( y = -2x/3 - 1/3 \)

close solution

Find the equation of the line, in slope-intercept form, with slope \( 2 \) passing through the point \( (-1, -6) \).

Problem Statement

Find the equation of the line, in slope-intercept form, with slope \( 2 \) passing through the point \( (-1, -6) \).

Final Answer

\( y = 2x - 4 \)

Problem Statement

Find the equation of the line, in slope-intercept form, with slope \( 2 \) passing through the point \( (-1, -6) \).

Solution

2711 video

Final Answer

\( y = 2x - 4 \)

close solution

Find the equation of the line, in slope-intercept form, with slope \( 3 \) passing through the point \( (-2, 3) \).

Problem Statement

Find the equation of the line, in slope-intercept form, with slope \( 3 \) passing through the point \( (-2, 3) \).

Final Answer

\( y = 3x + 9 \)

Problem Statement

Find the equation of the line, in slope-intercept form, with slope \( 3 \) passing through the point \( (-2, 3) \).

Solution

We are given the slope \( m = 3 \) and a point on the line at \( (-2, 3) \).

Let's use the point-slope form \( y-y_1 = m(x - x_1) \).

\( y-3 = 3( x - (-2)) \)

\( y - 3 = 3x + 6 \)

\( y = 3x + 6 + 3 \)

Final Answer: \( y = 3x + 9 \)

If we used the slope-intercept form to solve this problem, we would have this.

\( y = mx + b \to 3 = 3(-2) + b \)

\( b = 3 + 6 \to b = 9 \)

Final Answer: \( y = 3x + 9 \)

Final Answer

\( y = 3x + 9 \)

close solution

Even though your goal is to have the slope and one point on the line, you will not usually be given this exact information in order to find the equation of the line. Sometimes you are given two points. From that information you need to determine the slope and then use one of the points to get the equation of the line. These problems will give you practice with this.

Instructions - - Unless otherwise instructed, find the equation of the line that passes through the two given points. Give your answer in slope-intercept form.

Find the equation of the line in slope-intercept form that passes through \((-3,7)\) and \((5,-1)\).

Problem Statement

Find the equation of the line in slope-intercept form that passes through \((-3,7)\) and \((5,-1)\).

Final Answer

\(y=-x+4\)

Problem Statement

Find the equation of the line in slope-intercept form that passes through \((-3,7)\) and \((5,-1)\).

Solution

2712 video

Final Answer

\(y=-x+4\)

close solution

Find the equation of the line in slope-intercept form that passes through the points \((3,-2)\) and \((2,-1)\).

Problem Statement

Find the equation of the line in slope-intercept form that passes through the points \((3,-2)\) and \((2,-1)\).

Final Answer

\( y=-x+1 \)

Problem Statement

Find the equation of the line in slope-intercept form that passes through the points \((3,-2)\) and \((2,-1)\).

Solution

2713 video

Final Answer

\( y=-x+1 \)

close solution

Find the equation of the line in slope-intercept form that contains \( (5,0) \) and \( (-4,3) \).

Problem Statement

Find the equation of the line in slope-intercept form that contains \( (5,0) \) and \( (-4,3) \).

Final Answer

\( y = -x/3 + 5/3 \)

Problem Statement

Find the equation of the line in slope-intercept form that contains \( (5,0) \) and \( (-4,3) \).

Solution

2714 video

Final Answer

\( y = -x/3 + 5/3 \)

close solution

Determine the equation of the line passing through the points \( (1, 1) \) and \( (5, -1) \), giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line passing through the points \( (1, 1) \) and \( (5, -1) \), giving your answer in slope-intercept form.

Final Answer

\( y = -x/2 + 3/2 \)

Problem Statement

Determine the equation of the line passing through the points \( (1, 1) \) and \( (5, -1) \), giving your answer in slope-intercept form.

Solution

2715 video

Final Answer

\( y = -x/2 + 3/2 \)

close solution

Determine the equation of the line passing through the points \( (-1, 1) \) and \( (1, 7) \), giving your answer in standard form \( Ax + By = C \).

Problem Statement

Determine the equation of the line passing through the points \( (-1, 1) \) and \( (1, 7) \), giving your answer in standard form \( Ax + By = C \).

Final Answer

\( 3x - y = -4 \)

Problem Statement

Determine the equation of the line passing through the points \( (-1, 1) \) and \( (1, 7) \), giving your answer in standard form \( Ax + By = C \).

Solution

2716 video

Final Answer

\( 3x - y = -4 \)

close solution

Determine the equation of the line passing through \( (-2, -3) \) and \( (4, -2) \). Give your answer in slope-intercept form.

Problem Statement

Determine the equation of the line passing through \( (-2, -3) \) and \( (4, -2) \). Give your answer in slope-intercept form.

Final Answer

\( y = x/6 - 8/3 \)

Problem Statement

Determine the equation of the line passing through \( (-2, -3) \) and \( (4, -2) \). Give your answer in slope-intercept form.

Solution

2717 video

Final Answer

\( y = x/6 - 8/3 \)

close solution

Find the equation of the line that goes through \( (-3, 5) \) and \( (2, 8) \). Use the point-slope form \( y - y_1 = m(x - x_1) \) and the second point. Give your answer in slope-intercept form.

Problem Statement

Find the equation of the line that goes through \( (-3, 5) \) and \( (2, 8) \). Use the point-slope form \( y - y_1 = m(x - x_1) \) and the second point. Give your answer in slope-intercept form.

Final Answer

\( y = 3x/5+ 34/5 \)

Problem Statement

Find the equation of the line that goes through \( (-3, 5) \) and \( (2, 8) \). Use the point-slope form \( y - y_1 = m(x - x_1) \) and the second point. Give your answer in slope-intercept form.

Solution

2718 video

Final Answer

\( y = 3x/5+ 34/5 \)

close solution

Determine the equation of the line with an x-intercept of \(3\) and a y-intercept of \(-4\). Give your answer in slope-intercept form.

Problem Statement

Determine the equation of the line with an x-intercept of \(3\) and a y-intercept of \(-4\). Give your answer in slope-intercept form.

Final Answer

\( y = 4x/3 - 4 \)

Problem Statement

Determine the equation of the line with an x-intercept of \(3\) and a y-intercept of \(-4\). Give your answer in slope-intercept form.

Solution

2719 video

Final Answer

\( y = 4x/3 - 4 \)

close solution

Okay, before going on, here is great video on what we are going to discuss next, parallel and perpendicular lines.

Finding Equations of Parallel and Perpendicular Lines

Parallel Lines

When two lines are parallel, their slopes are equal. So, for two lines in slope intercept form \(y = mx + b_1\) and \( y=mx + b_2 \), the slopes \(m\) are the same but \( b_1 \neq b_2 \) (unless of course they are the same line). One of the nice things about the slope intercept form, \(y = mx + b\) is that you can pull the slope \(m\) and the y-intercept \(b\) directly from the equation without having to do any work. For the two parallel lines, the slopes are the same but they cross the y-axis at different points.

Instructions - - Unless otherwise instructed, find the equation of the line in slope-intercept parallel to the given line going through the given point.

Find the equation of the line in slope-intercept parallel to \( 3x-y/4 = 2 \) going through the point \((1/6,-8)\).

Problem Statement

Find the equation of the line in slope-intercept parallel to \( 3x-y/4 = 2 \) going through the point \((1/6,-8)\).

Final Answer

\(y=12x-10\)

Problem Statement

Find the equation of the line in slope-intercept parallel to \( 3x-y/4 = 2 \) going through the point \((1/6,-8)\).

Solution

2720 video

Final Answer

\(y=12x-10\)

close solution

Write the equation of the line that is parallel to \( x - 2y = 6 \) and passes through the point \( (-4, 3) \), giving your answer in slope-intercept form.

Problem Statement

Write the equation of the line that is parallel to \( x - 2y = 6 \) and passes through the point \( (-4, 3) \), giving your answer in slope-intercept form.

Final Answer

\( y = x/2 + 5 \)

Problem Statement

Write the equation of the line that is parallel to \( x - 2y = 6 \) and passes through the point \( (-4, 3) \), giving your answer in slope-intercept form.

Solution

2721 video

Final Answer

\( y = x/2 + 5 \)

close solution

Determine the equation of the line that is parallel to \( y = -5x + 4 \) and passes through \( (-1, 4) \), giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line that is parallel to \( y = -5x + 4 \) and passes through \( (-1, 4) \), giving your answer in slope-intercept form.

Final Answer

\( y = -5x - 1 \)

Problem Statement

Determine the equation of the line that is parallel to \( y = -5x + 4 \) and passes through \( (-1, 4) \), giving your answer in slope-intercept form.

Solution

2722 video

Final Answer

\( y = -5x - 1 \)

close solution

Determine the equation of the line that is parallel to \( 3x+2y = 12 \) and passes through \( (2, -5) \), giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line that is parallel to \( 3x+2y = 12 \) and passes through \( (2, -5) \), giving your answer in slope-intercept form.

Final Answer

\( y = -3x/2 + 6 \)

Problem Statement

Determine the equation of the line that is parallel to \( 3x+2y = 12 \) and passes through \( (2, -5) \), giving your answer in slope-intercept form.

Solution

2723 video

Final Answer

\( y = -3x/2 + 6 \)

close solution

Find the equation of the line parallel to the line \( y = 2x + 1 \) that passes through the point \( (1,5) \), giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line parallel to the line \( y = 2x + 1 \) that passes through the point \( (1,5) \), giving your answer in slope-intercept form.

Final Answer

\( y = 2x + 3 \)

Problem Statement

Find the equation of the line parallel to the line \( y = 2x + 1 \) that passes through the point \( (1,5) \), giving your answer in slope-intercept form.

Solution

2724 video

Final Answer

\( y = 2x + 3 \)

close solution

Find the equation of the line parallel to the line \( y = 3x - 4 \) that passes through the point \( (6, 4) \), giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line parallel to the line \( y = 3x - 4 \) that passes through the point \( (6, 4) \), giving your answer in slope-intercept form.

Final Answer

\( y = 3x - 14 \)

Problem Statement

Find the equation of the line parallel to the line \( y = 3x - 4 \) that passes through the point \( (6, 4) \), giving your answer in slope-intercept form.

Solution

2725 video

Final Answer

\( y = 3x - 14 \)

close solution

Find the equation of the line parallel to the line \( y = 3x/2 + 3 \) that passes through the point \( (4, -3) \), giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line parallel to the line \( y = 3x/2 + 3 \) that passes through the point \( (4, -3) \), giving your answer in slope-intercept form.

Final Answer

\( y = 3x/2 - 9 \)

Problem Statement

Find the equation of the line parallel to the line \( y = 3x/2 + 3 \) that passes through the point \( (4, -3) \), giving your answer in slope-intercept form.

Solution

2726 video

Final Answer

\( y = 3x/2 - 9 \)

close solution

Find the equation of the line in slope-intercept parallel to \( x+7y=4 \) through the point \((7,9)\).

Problem Statement

Find the equation of the line in slope-intercept parallel to \( x+7y=4 \) through the point \((7,9)\).

Final Answer

\( y=-x/7+10\)

Problem Statement

Find the equation of the line in slope-intercept parallel to \( x+7y=4 \) through the point \((7,9)\).

Solution

2727 video

Final Answer

\( y=-x/7+10\)

close solution

Perpendicular Lines

In the case of perpendicular lines, if the equation of one line is \(y = mx + b_1\), the equation of the other is \(y=-x/m+b_2\) So the slope of the second line is said to be the negative reciprocal of the first line.

Okay, time for the remaining practice problems.

Instructions - - Unless otherwise instructed, Find the equation of the line, in slope-intercept form, perpendicular to the given line going through the given point.

Find the equation of the line, in slope-intercept form, perpendicular to \( 9x+12y=2 \) containing the point \((15,7)\).

Problem Statement

Find the equation of the line, in slope-intercept form, perpendicular to \( 9x+12y=2 \) containing the point \((15,7)\).

Final Answer

\( y=4x/3 - 13 \)

Problem Statement

Find the equation of the line, in slope-intercept form, perpendicular to \( 9x+12y=2 \) containing the point \((15,7)\).

Solution

2728 video

Final Answer

\( y=4x/3 - 13 \)

close solution

Find the equation of the line, in slope-intercept form, perpendicular to \(10x+2y=9\) going through the point \((20,3)\).

Problem Statement

Find the equation of the line, in slope-intercept form, perpendicular to \(10x+2y=9\) going through the point \((20,3)\).

Final Answer

\( y = x/5 - 1 \)

Problem Statement

Find the equation of the line, in slope-intercept form, perpendicular to \(10x+2y=9\) going through the point \((20,3)\).

Solution

2729 video

Final Answer

\( y = x/5 - 1 \)

close solution

Determine the equation of the line that is perpendicular to \( x - 4y = 15 \) and passes through \( (3, 1) \), giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line that is perpendicular to \( x - 4y = 15 \) and passes through \( (3, 1) \), giving your answer in slope-intercept form.

Final Answer

\( y = -4x + 13 \)

Problem Statement

Determine the equation of the line that is perpendicular to \( x - 4y = 15 \) and passes through \( (3, 1) \), giving your answer in slope-intercept form.

Solution

2730 video

Final Answer

\( y = -4x + 13 \)

close solution

Write the equation of the line that contains the point \( (3, 5) \) and is perpendicular to the line \( y = -3x + 2 \), giving your answer in slope-intercept form.

Problem Statement

Write the equation of the line that contains the point \( (3, 5) \) and is perpendicular to the line \( y = -3x + 2 \), giving your answer in slope-intercept form.

Final Answer

\( y = x/3 + 4 \)

Problem Statement

Write the equation of the line that contains the point \( (3, 5) \) and is perpendicular to the line \( y = -3x + 2 \), giving your answer in slope-intercept form.

Solution

2731 video

Final Answer

\( y = x/3 + 4 \)

close solution

Determine the equation of the line that is perpendicular to \( y = 2x + 1 \) and passes through \( (-4, 5) \), giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line that is perpendicular to \( y = 2x + 1 \) and passes through \( (-4, 5) \), giving your answer in slope-intercept form.

Final Answer

\( y = -x/2 + 3 \)

Problem Statement

Determine the equation of the line that is perpendicular to \( y = 2x + 1 \) and passes through \( (-4, 5) \), giving your answer in slope-intercept form.

Solution

2732 video

Final Answer

\( y = -x/2 + 3 \)

close solution

Find the equation of the line perpendicular to the line \( y = 5x - 3 \) that passes through the point \( (2, -1) \), giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line perpendicular to the line \( y = 5x - 3 \) that passes through the point \( (2, -1) \), giving your answer in slope-intercept form.

Final Answer

\( y = -x/5 - 3/5 \)

Problem Statement

Find the equation of the line perpendicular to the line \( y = 5x - 3 \) that passes through the point \( (2, -1) \), giving your answer in slope-intercept form.

Solution

2733 video

Final Answer

\( y = -x/5 - 3/5 \)

close solution

Find the equation of the line perpendicular to the line \( y = 3x/4 - 1 \) that passes through the point \( (8, -3) \), giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line perpendicular to the line \( y = 3x/4 - 1 \) that passes through the point \( (8, -3) \), giving your answer in slope-intercept form.

Final Answer

\( y = -4x/3 + 23/3 \)

Problem Statement

Find the equation of the line perpendicular to the line \( y = 3x/4 - 1 \) that passes through the point \( (8, -3) \), giving your answer in slope-intercept form.

Solution

2734 video

Final Answer

\( y = -4x/3 + 23/3 \)

close solution

Find the equation of the line perpendicular to the line \( y = -2x/3 + 4 \) that passes through the point \( (6, -2) \), giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line perpendicular to the line \( y = -2x/3 + 4 \) that passes through the point \( (6, -2) \), giving your answer in slope-intercept form.

Final Answer

\( y = 3x/2 - 11 \)

Problem Statement

Find the equation of the line perpendicular to the line \( y = -2x/3 + 4 \) that passes through the point \( (6, -2) \), giving your answer in slope-intercept form.

Solution

2735 video

Final Answer

\( y = 3x/2 - 11 \)

close solution

Find the equation of the line in slope-intercept perpendicular to \( y=2x+1 \) through the point \((2,0)\).

Problem Statement

Find the equation of the line in slope-intercept perpendicular to \( y=2x+1 \) through the point \((2,0)\).

Final Answer

\( y = -x/2 + 1 \)

Problem Statement

Find the equation of the line in slope-intercept perpendicular to \( y=2x+1 \) through the point \((2,0)\).

Solution

2736 video

Final Answer

\( y = -x/2 + 1 \)

close solution

Find the equation of the line in slope-intercept perpendicular to \( y=-9x+5 \) through the point \((3,9)\).

Problem Statement

Find the equation of the line in slope-intercept perpendicular to \( y=-9x+5 \) through the point \((3,9)\).

Final Answer

\( y = x/9 + 26/3 \)

Problem Statement

Find the equation of the line in slope-intercept perpendicular to \( y=-9x+5 \) through the point \((3,9)\).

Solution

2737 video

Final Answer

\( y = x/9 + 26/3 \)

close solution

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