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Intermediate Geometry : How to find the equation of a line

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Example Questions

Example Question #21 : How To Find The Equation Of A Line

Find the equation of the line that goes through the points (11, 2) and (11, 10) .

Possible Answers:

x=12y+2

x=11

y=11x-10

y=11

Correct answer:

x=11

Explanation:

Recall that the slope-intercept form of a line:

y=mx+b ,

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

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

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

$\text{Slope}=\frac{10-2}{11-11}=\text{Undefined}$

Since the slope of the line is undefined, this must be just a vertical line with the equation x=11 .

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Example Question #22 : How To Find The Equation Of A Line

Find the equation of the line that passes through the points (5, 7) and (-8, 7) .

Possible Answers:

$x=\frac{13}{2}$

y=-x+14

$y=\frac{4}{5}x-\frac{3}{5}$

y=7

Correct answer:

y=7

Explanation:

Recall that the slope-intercept form of a line:

y=mx+b ,

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

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

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

$\text{Slope}=\frac{7-7}{5-(-8)}=0$

Since the slope is , the line is a horizontal line with the equation y=7 .

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Example Question #23 : How To Find The Equation Of A Line

Find the equation of the line that passes through the points (14, 9) and (-6, 12) .

Possible Answers:

$y=\frac{3}{26}x-\frac{113}{13}$

$y=-\frac{15}{26}x-\frac{111}{26}$

$y=-\frac{3}{20}x+\frac{111}{10}$

$y=\frac{20}{3}x+\frac{120}{11}$

Correct answer:

$y=-\frac{3}{20}x+\frac{111}{10}$

Explanation:

Recall that the slope-intercept form of a line:

y=mx+b ,

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

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

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

$\text{Slope}=\frac{12-9}{-6-14}=-\frac{3}{20}$

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

$y=-\frac{3}{20}x+b$

Plugging in (14, 9) yields the following:

$9=-\frac{3}{20}(14)+b$

Solve for .

$b-\frac{21}{10}=9$

$b=\frac{111}{10}$

The equation of the line is then $y=-\frac{3}{20}x+\frac{111}{10}$ .

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Example Question #24 : How To Find The Equation Of A Line

Find the equation of the line that passes through the points (20, -3) and (-45, 6) .

Possible Answers:

$y=-\frac{24}{13}x+\frac{4}{13}$

$y=-\frac{9}{65}x-\frac{3}{13}$

$y=-\frac{13}{24}x-\frac{1}{12}$

$y=-\frac{65}{4}x-\frac{5}{2}$

Correct answer:

$y=-\frac{9}{65}x-\frac{3}{13}$

Explanation:

Recall that the slope-intercept form of a line:

y=mx+b ,

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

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

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

$\text{Slope}=\frac{6-(-3)}{-45-20}=-\frac{9}{65}$

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

$y=-\frac{9}{65}x+b$

Plugging in (20, -3) yields the following:

$-3=-\frac{9}{65}(20)+b$

Solve for .

$b-\frac{36}{13}=-3$

$b=-\frac{3}{13}$

The equation of the line is then $y=-\frac{9}{65}x-\frac{3}{13}$ .

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Example Question #25 : How To Find The Equation Of A Line

Find the equation of the line that passes through the points (-17, 1) and (-1, 10) .

Possible Answers:

$y=-\frac{9}{16}x+\frac{3}{16}$

$y=\frac{9}{4}x+\frac{107}{4}$

$y=\frac{16}{9}x+\frac{100}{9}$

$y=\frac{9}{16}x+\frac{169}{16}$

Correct answer:

$y=\frac{9}{16}x+\frac{169}{16}$

Explanation:

Start by finding the slope of the line.

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

$\text{Slope}=\frac{10-1}{-1-(-17)}=\frac{9}{16}$

Now, take one of the points and put the equation into point-slope form. Recall the point-slope form of an equation of a line:

y-y_1=m(x-x_1) , where is the slope and (x_1, y_1) is the point.

$y-10=\frac{9}{16}(x+1)$

Now, simplify this equation so that it is in slope-intercept form. Recall the slope-intercept form of an equation of a line:

y=mx+b , where is the slope and is the y-intercept.

$y-10=\frac{9}{16}(x+1)$

$y-10=\frac{9}{16}x+\frac{9}{16}$

$y=\frac{9}{16}x+\frac{169}{16}$

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Example Question #26 : How To Find The Equation Of A Line

Find the equation of a line that passes through the points (4, 12) and (12, 10) .

Possible Answers:

y=4x+10

$y=\frac{1}{4}x+16$

y=-4x+8

$y=-\frac{1}{4}x+13$

Correct answer:

$y=-\frac{1}{4}x+13$

Explanation:

Start by finding the slope of the line.

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

$\text{Slope}=\frac{10-12}{12-4}=-\frac{1}{4}$

Now, take one of the points and put the equation into point-slope form. Recall the point-slope form of an equation of a line:

y-y_1=m(x-x_1) , where is the slope and (x_1, y_1) is the point.

$y-12=-\frac{1}{4}(x-4)$

Now, simplify this equation so that it is in slope-intercept form. Recall the slope-intercept form of an equation of a line:

y=mx+b , where is the slope and is the y-intercept.

$y-12=-\frac{1}{4}(x-4)$

$y-12=-\frac{1}{4}x+1$

$y=-\frac{1}{4}x+13$

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Example Question #27 : How To Find The Equation Of A Line

Find the equation of the line that passes through the points $(-\frac{1}{2}, 6)$ and $(\frac{3}{2}, 7)$ .

Possible Answers:

$y=\frac{1}{2}x+\frac{21}{2}$

$y=\frac{1}{2}x+\frac{9}{2}$

$y=\frac{1}{2}x+\frac{25}{4}$

$y=\frac{1}{2}x+\frac{19}{4}$

Correct answer:

$y=\frac{1}{2}x+\frac{25}{4}$

Explanation:

Start by finding the slope of the line.

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

$\text{Slope}=\frac{7-6}{\frac{3}{2}-(-\frac{1}{2})}=\frac{1}{2}$

Now, take one of the points and put the equation into point-slope form. Recall the point-slope form of an equation of a line:

y-y_1=m(x-x_1) , where is the slope and (x_1, y_1) is the point.

$y-7=\frac{1}{2}(x-\frac{3}{2})$

Now, simplify this equation so that it is in slope-intercept form. Recall the slope-intercept form of an equation of a line:

y=mx+b , where is the slope and is the y-intercept.

$y-7=\frac{1}{2}(x-\frac{3}{2})$

$y-7=\frac{1}{2}x-\frac{3}{4}$

$y-7=\frac{1}{2}x+\frac{25}{4}$

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Example Question #54 : Other Lines

Find the equation of the line that passes through the points (-21, 6) and (10, -5) .

Possible Answers:

$y=-\frac{11}{31}x-\frac{47}{31}$

$y=-\frac{11}{31}x-\frac{45}{31}$

$y=-\frac{11}{31}x-\frac{31}{11}$

$y=-\frac{11}{31}x-\frac{41}{11}$

Correct answer:

$y=-\frac{11}{31}x-\frac{45}{31}$

Explanation:

Start by finding the slope of the line.

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

$\text{Slope}=\frac{-5-6}{10-(-21)}=-\frac{11}{31}$

Now, take one of the points and put the equation into point-slope form. Recall the point-slope form of an equation of a line:

y-y_1=m(x-x_1) , where is the slope and (x_1, y_1) is the point.

$y+5=-\frac{11}{31}(x-10)$

Now, simplify this equation so that it is in slope-intercept form. Recall the slope-intercept form of an equation of a line:

y=mx+b , where is the slope and is the y-intercept.

$y+5=-\frac{11}{31}(x-10)$

$y+5=-\frac{11}{31}x+\frac{110}{31}$

$y=-\frac{11}{31}x-\frac{45}{31}$

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Example Question #55 : Other Lines

Find the equation of the line that passes through the points (16, 16) and (-8, 10) .

Possible Answers:

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

$y=\frac{1}{4}x+10$

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

$y=\frac{1}{4}x+12$

Correct answer:

$y=\frac{1}{4}x+12$

Explanation:

Start by finding the slope of the line.

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

$\text{Slope}=\frac{10-16}{-8-16}=\frac{1}{4}$

Now, take one of the points and put the equation into point-slope form. Recall the point-slope form of an equation of a line:

y-y_1=m(x-x_1) , where is the slope and (x_1, y_1) is the point.

$y-16=\frac{1}{4}(x-16)$

Now, simplify this equation so that it is in slope-intercept form. Recall the slope-intercept form of an equation of a line:

y=mx+b , where is the slope and is the y-intercept.

$y-16=\frac{1}{4}(x-16)$

$y-16=\frac{1}{4}x-4$

$y=\frac{1}{4}x+12$

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Example Question #56 : Other Lines

Find the equation of the line that passes through the points (-5, 7) and (9, 6) .

Possible Answers:

$y=-\frac{1}{14}x+\frac{46}{7}$

$y=-\frac{1}{14}x+\frac{93}{14}$

$y=-\frac{1}{14}x-\frac{23}{14}$

$y=-\frac{1}{4}x+\frac{15}{14}$

Correct answer:

$y=-\frac{1}{14}x+\frac{93}{14}$

Explanation:

Start by finding the slope of the line.

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

$\text{Slope}=\frac{6-7}{9-(-5)}=-\frac{1}{14}$

Now, take one of the points and put the equation into point-slope form. Recall the point-slope form of an equation of a line:

y-y_1=m(x-x_1) , where is the slope and (x_1, y_1) is the point.

$y-6=-\frac{1}{14}(x-9)$

Now, simplify this equation so that it is in slope-intercept form. Recall the slope-intercept form of an equation of a line:

y=mx+b , where is the slope and is the y-intercept.

$y-6=-\frac{1}{14}(x-9)$

$y-6=-\frac{1}{14}x+\frac{9}{14}$

$y=-\frac{1}{14}x+\frac{93}{14}$

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Intermediate Geometry : How to find the equation of a line

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #21 : How To Find The Equation Of A Line

Example Question #22 : How To Find The Equation Of A Line

Example Question #23 : How To Find The Equation Of A Line

Example Question #24 : How To Find The Equation Of A Line

Example Question #25 : How To Find The Equation Of A Line

Example Question #26 : How To Find The Equation Of A Line

Example Question #27 : How To Find The Equation Of A Line

Example Question #54 : Other Lines

Example Question #55 : Other Lines

Example Question #56 : Other Lines

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