How to find the equation of a line - Intermediate Geometry

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Question

Given two points $\left ( -3,-3 \right )$ and $\left ( -1,1 \right )$ , find the equation for the line connecting those two points in slope-intercept form.

Answer

If we have two points, we can find the slope of the line between them by using the definition of the slope:

$slope = \frac{\Delta Y}{\Delta X}$ where the triangle is the greek letter 'Delta', and is used as a symbol for 'difference' or 'change in'

Now that we have our slope ( $\frac{4}{2}$ , simplified to ), we can write the equation for slope-intercept form:

where is the slope and is the y-intercept

In order to find the y-intercept, we simply plug in one of the points on our line

So our equation looks like

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Question

Which of the following is an equation for a line with a slope of and a y-intercept of ?

Answer

Because we have the desired slope and the y-intercept, we can easily write this as an equation in slope-intercept form (y=mx+b).

This gives us . Because this does not match either of the answers in this form (y=mx+b), we must solve the equation for x. Adding 5 to each side gives us . We can then multiple both sides by 3 and divide both sides by 4, giving us .

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Question

If the -intercept of a line is , and the -intercept is , what is the equation of this line?

Answer

If the y-intercept of a line is , then the -value is when is zero. Write the point:

If the -intercept of a line is , then the -value is when is zero. Write the point:

Use the following formula for slope and the two points to determine the slope:

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{0-5}{-1-0}= \frac{-5}{-1} =5$

Use the slope intercept form and one of the points, suppose , to find the equation of the line by substituting in the values of the point and solving for , the -intercept.

Therefore, the equation of this line is .

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Question

A line goes through the following points and .

Find the equation of the line.

Answer

First, find the slope of the line using the slope formula:

$\frac{y_2-y_1}{x_2-x_1}=\frac{7-3}{3-2}=\frac{4}{1}=4$ .

Next we plug one of the points, and the slope, into the point-intercept line forumula:

where m is our slope.

Then and when we plug in point (2,3) the formula reads then solve for b.

.

To find the equation of the line, we plug in our m and b into the slope-intercept equation.

So, or simplified, .

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Question

What is the equation of a line that has a slope of and a -intercept of ?

Answer

The slope intercept form can be written as:

where is the slope and is the y-intercept. Plug in the values of the slope and -intercept into the equation.

The correct answer is:

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Question

What is the equation of a line with a slope of and an -intercept of ?

Answer

The -intercept is the value of when the value is equal to zero. The actual point located on the graph for an -intercept of is . The slope, , is 2.

Write the slope-intercept equation and substitute the point and slope to solve for the -intercept:

Plug the slope and -intercept back in the slope-intercept formula:

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Question

Write the equation for the line passing through the points and

Answer

To determine the equation, first find the slope:

$\frac{\Delta y}{\Delta x} = \frac{3--1}{-4-4} = \frac{4 }{-8 } =- \frac{1}{2}$

We want this equation in slope-intercept form, . We know and because we have two coordinate pairs to choose from representing an and a . We know because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose :

$-1 = -\frac{1}{2} (4) + b$ multiply ""

add to both sides

This means that the form is $y = -\frac{1}{2}x +1$

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Question

Write the equation for a line that passes through the points and .

Answer

To determine the equation, first find the slope:

$\frac{\Delta y}{\Delta x} = \frac{3-2}{2--1} = \frac{1 }{3 }$

We want this equation in slope-intercept form, . We know and because we have two coordinate pairs to choose from representing an and a . We know because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose :

$2= \frac{1}{3} (3) + b$ multiply ""

subtract from both sides

This means that the form is $y = \frac{1}{3}x +1$

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Question

Find the equation for a line passing through the points and .

Answer

To determine the equation, first find the slope:

$\frac{\Delta y}{\Delta x} = \frac{-7--1}{3-5} = \frac{-6 }{-2 } =3$

We want this equation in slope-intercept form, . We know and because we have two coordinate pairs to choose from representing an and a . We know because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose :

multiply ""

subtract from both sides

This means that the form is

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Question

Find the equation for the line passing through the points and .

Answer

To determine the equation, first find the slope:

$\frac{\Delta y}{\Delta x} = \frac{2--4}{6--3} = \frac{6}{9 } =\frac{2}{3}$

We want this equation in slope-intercept form, . We know and because we have two coordinate pairs to choose from representing an and a . We know because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose :

$2 = \frac{2}{3} (6) + b$ multiply ""

subtract from both sides

This means that the form is $y = \frac{2}{3}x-2$

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Question

Write the equation for the line passing through the points and .

Answer

First, find the slope of the line:

$\frac{ \Delta y }{ \Delta x } = \frac{2 - 1 }{ 4 - - 2 } = \frac{ 1 }{ 6 }$

Now we want to find the y-intercept. We can figure this out by plugging in the slope for "m" and one of the points in for x and y in the formula :

$1 = \frac{1}{6} (-2 ) + b$

$1 = -\frac{1}{3} + b$

$1 \frac{1}{3} = b$

The equation is $y = \frac{ 1}{6} x + 1 \frac{1}{3}$

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Question

Find the equation for the line passing through the points and .

Answer

First, determine the slope of the line using the slope formula:

$\frac{ \Delta y }{ \Delta x } = \frac{ y_2 - y_1 }{ x _ 2 - x_1 } = \frac{7 - 3} { 5 - 2 } = \frac{4}{3}$

The equation will be in the form where m is the slope that we just determined, and b is the y-intercept. To determine that, we can plug in the slope for m and the coordinates of one of the original points for x and y:

$3 = \frac{4}{3} (2) + b$

$3 = \frac{8 }{3 } + b$ to subtract, it will be easier to convert 3 to a fraction, $\frac{9}{3}$

$\frac{9}{3} - \frac{8}{3} = b$

$\frac{1}{3} = b$

The equation is $y = \frac{4}{3} x + \frac{1}{3}$

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Question

Find the equation of a line passing through the points and .

Answer

To find the equation of a line passing through these points we must find a line with that same slope. Start by finding the slope between the two points and then use the point slope equation to find the equation of the line.

slope:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{0--5}{5-0}=1$

Now use the point slope equation:

$y-y_{1}=m(x-x_{1})\rightarrow y-(-5)=1(x-0)\rightarrow \mathbf{y=x-5}$

*make sure you use the SAME coordinate pair when substituting x and y into the point slope equation.

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Question

Find the equation of a line that passes through the following points:

and

Answer

Recall that the the following is the slope-intercept form of a line:

In this equation, the variables are represented by the following:

$m=\text{slope}$

$b=\text{y-intercept}$

Find the slope of the line by using the following formula:

$\text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}$

In this equation, the x- and y-variables correspond to the coordinates of the given points.

$\text{Slope}=\frac{12-8}{6-5}$

$\text{Slope}=\frac{4}{1}$

$\text{Slope}=4$

Next, find the y-intercept of the line by substituting one of the points into the semi-completed formula.

Substituting in the point yields the following:

Rearrange and solve for .

Subtract 20 from both sides of the equation.

Substitute this value of the y-intercept into our semi-complete equation to get the answer:

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Question

Find the equation of a line that goes through the points and .

Answer

Recall that the slope-intercept form of a line:

,

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

$\text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{5-12}{-2-3}=\frac{7}{5}$

Next, find the y-intercept of the line by plugging in of the points into the semi-completed formula.

$y=\frac{7}{5}x+b$

Plugging in yields the following:

$12=\frac{7}{5}(3)+b$

Solve for .

$b+\frac{21}{5}=12$

$b=\frac{39}{5}$

The equation of the line is then $y=\frac{7}{5}x+\frac{39}{5}$ .

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Question

Find the equation of a line that goes through the points and .

Answer

Recall that the slope-intercept form of a line:

,

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

$\text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{8-7}{12-(-10)}=\frac{1}{22}$

Next, find the y-intercept of the line by plugging in of the points into the semi-completed formula.

$y=\frac{1}{22}x+b$

Plugging in yields the following:

$8=\frac{1}{22}(12)+b$

Solve for .

$b+\frac{6}{11}=8$

$b=\frac{82}{11}$

The equation of the line is then $y=\frac{1}{22}x+\frac{82}{11}$ .

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Question

Find the equation of a line that goes through the points and .

Answer

Recall that the slope-intercept form of a line:

,

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

$\text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{10-2}{16-15}=8$

Next, find the y-intercept of the line by plugging in of the points into the semi-completed formula.

Plugging in yields the following:

Solve for .

The equation of the line is then .

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Question

Find the equation of a line that goes through the points and .

Answer

Recall that the slope-intercept form of a line:

,

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

$\text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{8-(-5)}{-4-(-3)}=-13$

Next, find the y-intercept of the line by plugging in of the points into the semi-completed formula.

Plugging in yields the following:

Solve for .

The equation of the line is then .

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Question

Find the equation of the line that goes through the points and .

Answer

Recall that the slope-intercept form of a line:

,

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

$\text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{0-16}{12-0}=-\frac{4}{3}$

The y-intercept is since that is given as one of the points on the line.

The line must have the equation $y=-\frac{4}{3}x+16$ .

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Question

Find the equation of the line that goes through the points and .

Answer

Recall that the slope-intercept form of a line:

,

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

$\text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{18-(-36)}{20-(-40)}=\frac{9}{10}$

Next, find the y-intercept of the line by plugging in of the points into the semi-completed formula.

$y=\frac{9}{10}x+b$

Plugging in yields the following:

$18=\frac{9}{10}(20)+b$

Solve for .

The equation of the line is then $y=\frac{9}{10}x$ .

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