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

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

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What is the slope of the line with the equation

$-\frac{2}{3}$

$-\frac{3}{2}$

Explanation

To find the slope, put the equation in the form of .

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

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

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

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

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

Explanation

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

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:

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

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

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:

Therefore, the equation of this line is,

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

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

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

Explanation

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

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:

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

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

Now that we've found the slope of our line, $\frac{1}{3}$ , 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:

Therefore, the equation of this line is,

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

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

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

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

Explanation

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

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:

Therefore, the equation of this line is,

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

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

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

The line has no -intercept.

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 :

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

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

The -intercept is the point .

The equation of a line is . Find the slope of this line.

$\frac{3}{2}$

$-\frac{3}{2}$

$\frac{3}{4}$

$-\frac{3}{4}$

Explanation

To find the slope, you will need to put the equation in form. The value of will be the slope.

Subtract from either side:

Divide each side by :

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

You can now easily identify the value of .

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

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

Explanation

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

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:

Therefore, the equation of this line is,

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

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

Explanation

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

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:

Therefore, the equation of this line is,

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

A line has the equation . What is the slope of this line?

$\frac{5}{9}$

$-\frac{4}{3}$

$\frac{9}{5}$

Explanation

You need to put the equation in form before you can easily find out its slope.

$y=\frac{5}{9}x-\frac{4}{3}$

Since $m=\frac{5}{9}$ , that must be the slope.

Give the -intercept, if there is one, of the graph of the equation

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

The graph has no -intercept.

$\left ( 0, \frac{3}{2}\right )$

Explanation

The -intercept is the point at which the graph crosses the -axis; at this point, the -coordinate is 0, so substitute for in the equation:

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

$y = \frac{1}{0^{2}+0} + 3$

$y = \frac{1}{0} + 3$

However, since this expression has 0 in a denominator, it is of undefined value. This means that there is no value of paired with -coordinate 0, and, subsequently, the graph of the equation has no -intercept.

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