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Common Core: 8th Grade Math : Use Similar Triangles to Show Equal Slopes: CCSS.Math.Content.8.EE.B.6

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All Common Core: 8th Grade Math Resources

7 Diagnostic Tests 75 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #8 : How To Find X Or Y Intercept

Give the -intercept of the line that passes through points (-3, 4) and (2, -3) .

Possible Answers:

$\left (- \frac{1}{7} , 0 \right )$

$\left ( - \frac{1}{5},0 \right )$

$\left ( - 8 \frac{1}{5},0 \right )$

$\left (- 5\frac{6}{7} , 0 \right )$

$\text{The line has no } x\text{-intercept.}$

Correct answer:

$\left (- \frac{1}{7} , 0 \right )$

Explanation:

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

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

setting $x_{1} = -3, y_{1} = 4, x_{2} = 2, y_{2} = -3$ :

$m = \frac{-3 -4}{2-(-3)} = \frac{-7}{5} =- \frac{7}{5}$

By the point-slope formula, this line has the equation

$y - y_{1} = m(x -x_{1})$

where

$x_{1} = -3, y_{1} = 4, m = - \frac{7}{5}$ ; the line becomes

$y - 4 = - \frac{7}{5}(x -(-3))$

$y - 4 = - \frac{7}{5}(x +3)$

To find the -intercept, substitute 0 for and solve for :

$0 - 4 = - \frac{7}{5}(x +3)$

$- 4 = - \frac{7}{5} x - \frac{21}{5}$

$- 4+ \frac{21}{5} = - \frac{7}{5} x - \frac{21}{5}+ \frac{21}{5}$

$\frac{1}{5} = - \frac{7}{5} x$

$- \frac{5}{7} \cdot \frac{1}{5} = - \frac{5}{7}\left ( - \frac{7}{5} x \right )$

$- \frac{1}{7} =x$

The -intercept is $\left (- \frac{1}{7} , 0 \right )$ .

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Example Question #221 : Coordinate Geometry

Give the -intercept of the line with slope $\frac{2}{5}$ that passes through point (5, 9) .

Possible Answers:

$\left ( -27 \frac{1}{2}, 0 \right )$

$\text{The line has no } x \text{-intercept.}$

$\left ( -17 \frac{1}{2}, 0 \right )$

(-11, 0)

(7, 0)

Correct answer:

$\left ( -17 \frac{1}{2}, 0 \right )$

Explanation:

By the point-slope formula, this line has the equation

$y - y_{1} = m(x -x_{1})$

where

$x_{1} = 5, y_{1} = 9 ,m = \frac{2}{5}$

By substitution, the equation becomes

$y -9 = \frac{2}{5}(x -5)$

To find the -intercept, substitute 0 for and solve for :

$0-9 = \frac{2}{5}(x -5)$

$-9 = \frac{2}{5}x -2$

$-9 +2= \frac{2}{5}x -2+2$

$-7 = \frac{2}{5}x$

$\frac{5}{2} \cdot \left (-7 \right ) = \frac{5}{2} \cdot \frac{2}{5}x$

$x = -\frac{35}{2} = -17 \frac{1}{2}$

The -intercept is $\left ( -17 \frac{1}{2}, 0 \right )$ .

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Example Question #222 : Coordinate Geometry

Find the y-intercept: -y-8=-6x+2

Possible Answers:

Correct answer:

Explanation:

Rewrite the equation in slope-intercept form, y=mx+b .

-y-8=-6x+2

-y=-6x+10

y=6x-10

The y-intercept is , which is .

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Example Question #131 : Grade 8

What is the -intercept of the graph of the function

$f(x) = x^{2} + 3x - 10?$

Possible Answers:

$\left (0, 3 \frac{1}{3} \right )$

The graph has no -intercept.

$\left (0, 3 \right )$

$\left (0, -10 \right )$

$\left (0, 12 \frac{1}{4} \right )$

Correct answer:

$\left (0, -10 \right )$

Explanation:

The -intercept of the graph of a function is the point at which it intersects the -axis - that is, at which x = 0 . This point is $\left (0, f(0) \right )$ , so evaluate f(0) :

$f(x) = x^{2} + 3x - 10$

$f( 0) = 0^{2} + 3 \cdot 0 - 10 = 0 + 0 - 10 = -10$

The -intercept is $\left (0, -10 \right )$ .

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Example Question #2 : Slope

What is the slope of the line with the equation 2x+3y=-9?

Possible Answers:

$-\frac{3}{2}$

$-\frac{2}{3}$

Correct answer:

$-\frac{2}{3}$

Explanation:

To find the slope, put the equation in the form of y=mx+b .

2x+3y=-9

3y=-2x-9

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

Since $m=-\frac{2}{3}$ , that is the value of the slope.

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Example Question #101 : Expressions & Equations

Using the similar triangles, find the equation of the line in the provided graph.

Possible Answers:

y=3x+4

y=4x+4

y=2x+4

y=6x+4

Correct answer:

y=2x+4

Explanation:

The equation for a line can be written in the slope-intercept form:

y=mx+b ,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangles should have the same slope:

$\frac{4}{2}=2$

$\frac{6}{3}=2$

Now that we've found the slope of our line, , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

(0,4)

Therefore, the equation of this line is,

y=2x+4

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Example Question #141 : Grade 8

Using the similar triangles, find the equation of the line in the provided graph.

Possible Answers:

y=3x+2

y=2x+1

y=x+2

y=7x+2

Correct answer:

y=x+2

Explanation:

The equation for a line can be written in the slope-intercept form:

y=mx+b ,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangle should have the same slope:

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

$\frac{7}{7}=1$

Now that we've found the slope of our line, , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

(0,2)

Therefore, the equation of this line is,

y=x+2

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Example Question #103 : Expressions & Equations

Using the similar triangles, find the equation of the line in the provided graph.

Possible Answers:

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

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

y=2x+0

y=4x

Correct answer:

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

Explanation:

The equation for a line can be written in the slope-intercept form:

y=mx+b ,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangle should have the same slope:

$\frac{2}{4}=\frac{1}{2}$

$\frac{3}{6}=\frac{1}{2}$

Now that we've found the slope of our line, $\frac{1}{2}$ , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

(0,0)

Therefore, the equation of this line is,

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

Report an Error

Example Question #142 : Grade 8

Using the similar triangles, find the equation of the line in the provided graph.

Possible Answers:

y=x+1

y=8x+1

y=2x

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

Correct answer:

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

Explanation:

The equation for a line can be written in the slope-intercept form:

y=mx+b ,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangle should have the same slope:

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

$\frac{4}{8}=\frac{1}{2}$

Now that we've found the slope of our line, $\frac{1}{2}$ , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

(0,1)

Therefore, the equation of this line is,

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

Report an Error

Example Question #106 : Expressions & Equations

Using the similar triangles, find the equation of the line in the provided graph.

Possible Answers:

y=3x+7

$y=\frac{1}{3}x+7$

y=2x+7

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

Correct answer:

y=2x+7

Explanation:

The equation for a line can be written in the slope-intercept form:

y=mx+b ,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangle should have the same slope:

$\frac{6}{3}=2$

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

Now that we've found the slope of our line, , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

(0,7)

Therefore, the equation of this line is,

y=2x+7

Report an Error

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Common Core: 8th Grade Math : Use Similar Triangles to Show Equal Slopes: CCSS.Math.Content.8.EE.B.6

Study concepts, example questions & explanations for Common Core: 8th Grade Math

All Common Core: 8th Grade Math Resources

Example Questions

Example Question #8 : How To Find X Or Y Intercept

Example Question #221 : Coordinate Geometry

Example Question #222 : Coordinate Geometry

Example Question #131 : Grade 8

Example Question #2 : Slope

Example Question #101 : Expressions & Equations

Example Question #141 : Grade 8

Example Question #103 : Expressions & Equations

Example Question #142 : Grade 8

Example Question #106 : Expressions & Equations

All Common Core: 8th Grade Math Resources

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