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Intermediate Geometry : Other Lines

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Example Question #5 : How To Find The Equation Of A Line

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

Possible Answers:

y=2x-4

y=2x+2

y=2x+4

y=2x-2

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

Correct answer:

y=2x-4

Explanation:

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 (2,0) . The slope, , is 2.

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

y=mx+b

0=2(2)+b

b=-4

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

y=2x-4

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Example Question #6 : How To Find The Equation Of A Line

A line goes through the following points (2,3) and (3, 7) .

Find the equation of the line.

Possible Answers:

y=4x+13

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

y=4x+5

$y=\frac{1}{4}x+5$

y=4x-5

Correct answer:

y=4x-5

Explanation:

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:

y=mx+b where m is our slope.

Then y=4x+b and when we plug in point (2,3) the formula reads 3=4(2)+b then solve for b.

3-8=-5 =b .

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

So, y=4x+(-5) or simplified, y=4x-5 .

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Example Question #7 : How To Find The Equation Of A Line

Write the equation for the line passing through the points (4,-1) and (-4, 3)

Possible Answers:

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

$y = \frac{1}{2} x + 5$

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

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

$y = -\frac{1}{2}x + 5$

Correct answer:

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

Explanation:

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, y = mx+b . 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 (4, -1) :

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

-1 = -2 + b add to both sides

1 = b

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

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Example Question #8 : How To Find The Equation Of A Line

Write the equation for a line that passes through the points (3,2) and (2,-1) .

Possible Answers:

y = x-3

y = x + 1

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

$y = \frac{1}{3}x - 13$

y = 5x -13

Correct answer:

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

Explanation:

To determine the equation, first find the slope:

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

We can choose either point and get the correct answer. Let's choose (3,2) :

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

2 = 1 + b subtract from both sides

1 = b

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

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Example Question #9 : How To Find The Equation Of A Line

Find the equation for a line passing through the points (5, -1) and (3,-7) .

Possible Answers:

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

y = 3x-16

y = -3x +14

y = 3x + 2

$y = \frac{1}{3}x -6$

Correct answer:

y = 3x-16

Explanation:

To determine the equation, first find the slope:

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

We can choose either point and get the correct answer. Let's choose (3, -7) :

-7 = 3 (3) + b multiply ""

-7 =9+ b subtract from both sides

-16 = b

This means that the y = mx+b form is y = 3x -16

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Example Question #10 : How To Find The Equation Of A Line

Find the equation for the line passing through the points (6,2) and (-3,-4) .

Possible Answers:

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

y = 2x - 2

$y =- \frac{2}{3}x + 6$

y = 2x+2

$y = \frac{2}{3} x + 2$

Correct answer:

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

Explanation:

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 can choose either point and get the correct answer. Let's choose (6, 2) :

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

2 = 4 + b subtract from both sides

-2 = b

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

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Example Question #31 : Other Lines

Find the equation for the line passing through the points (5, 7) and (2, 3 ) .

Possible Answers:

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

$y = \frac{10}{3} x - \frac{11}{3}$

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

$y = \frac{3}{4} x + \frac{3}{2}$

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

Correct answer:

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

Explanation:

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 y = mx + b 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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Example Question #32 : Other Lines

Write the equation for the line passing through the points (4, 2) and (-2,1) .

Possible Answers:

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

y = x + 3

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

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

$y = \frac{1}{6} x + \frac{2}{3}$

Correct answer:

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

Explanation:

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 y = mx+b :

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

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

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

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

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Example Question #33 : Other Lines

Find the equation of a line passing through the points (0,-5) and (5,0) .

Possible Answers:

y=x-5

None of these.

y=-x+5

y=x+5

y=-x-5

Correct answer:

y=x-5

Explanation:

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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Example Question #34 : Other Lines

Find the equation of a line that goes through the points (3, 12) and (-2, 5) .

Possible Answers:

$y=\frac{7}{5}x+\frac{39}{5}$

$y=-\frac{1}{6}x-\frac{6}{5}$

$y=-\frac{2}{5}x-\frac{11}{5}$

$y=\frac{2}{3}x+\frac{4}{9}$

Correct answer:

$y=\frac{7}{5}x+\frac{39}{5}$

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:

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

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

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 (3, 12) 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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Intermediate Geometry : Other Lines

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #5 : How To Find The Equation Of A Line

Example Question #6 : How To Find The Equation Of A Line

Example Question #7 : How To Find The Equation Of A Line

Example Question #8 : How To Find The Equation Of A Line

Example Question #9 : How To Find The Equation Of A Line

Example Question #10 : How To Find The Equation Of A Line

Example Question #31 : Other Lines

Example Question #32 : Other Lines

Example Question #33 : Other Lines

Example Question #34 : Other Lines

All Intermediate Geometry Resources

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