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17Calculus Precalculus - Equations of Lines

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Single Variable Calculus
Derivatives
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Multi-Variable Calculus
Precalculus
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Equations of Lines

To describe the graph of a line, there are several equations that you will need. We summarize them in this table.

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.

FAQ: Why Is The Slope-Intercept Form Important?

We prefer the slope-intercept form as a final answer since all linear functions can be described by this equation. It is also easy to pull off these two important pieces of information directly from this equation.
1. The slope \(m\), which is the constant in front of the x-term, and
2. The point at which the line crosses the y-axis, \((0, b)\), also called the y-intercept.

Notice in the last paragraph that we used the term 'linear function.' This use of the word 'function' is deliberate. The only line that cannot be written in slope-intercept form is a vertical line. However, a vertical line is not a function since it does not pass the vertical line test.

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

Check out this GeoGebra tool, which will help you get a visualization of how changing the slope or the y-intercept of a line affects how the graph looks.

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.

Practice - Given Slope and One Point

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

slope: \( -2/3 \) 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

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

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slope: \( 3 \) 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

\( y = 3x + 9 \)

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slope: \( -2/3 \) 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 solution

Final Answer

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

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slope: \( 2 \) 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 solution

Final Answer

\( y = 2x - 4 \)

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Even though your goal is to have the slope and one point on the line, you may not 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.

Okay, let's work a few more practice problems.

Practice - Given Two Points

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

\((-3,7)\) 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 solution

Final Answer

\(y=-x+4\)

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\((3,-2)\) 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 solution

Final Answer

\( y=-x+1 \)

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\( (5,0) \) 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 solution

Final Answer

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

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\( (1, 1) \) and \( (5, -1) \)

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 solution

Final Answer

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

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\( (-1, 1) \) and \( (1, 7) \)

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 solution

Final Answer

\( 3x - y = -4 \)

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\( (-2, -3) \) and \( (4, -2) \)

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 solution

Final Answer

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

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\( (-3, 5) \) and \( (2, 8) \)

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 solution

Final Answer

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

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x-intercept: \(3\);     y-intercept: \(-4\)

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.

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 solution

Final Answer

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

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Practice Instructions

Unless otherwise instructed, find the equation of the line using the given data, giving your answer in slope-intercept form.

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