Linear Algebra - GED Math

Card 0 of 20

Question

Find the slope and y-intercept of the line depicted by the equation:
$\small y=-\frac{1}{3}x+7$

Answer

The equation is written in slope-intercept form, which is:
$\small y=mx+b$

where is equal to the slope and is equal to the y-intercept. Therefore, a line depicted by the equation

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

has a slope that is equal to $-\frac{1}{3}$ and a y-intercept that is equal to .

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Question

Find the slope and y-intercept of the line that is represented by the equation $y=\frac{2}{3}x-8$

Answer

The slope-intercept form of a line is: , where is the slope and is the y-intercept.

In this equation, $m \textup{ = }\frac{2}{3}$ and $y \textup{ = }-8$

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Question

What is the slope and y-intercept of the following line?

Answer

Convert the equation into slope-intercept form, which is , where is the slope and is the y-intercept.

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

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

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

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Question

Refer to above red line. What is its slope?

Answer

Given two points, $(x_{1}, y_{1}), (x_{2}, y_{2})$ , the slope can be calculated using the following formula:

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

Set $x_{1}=-4, y_{1}=x_{2}= 0, y_{2}=8$ :

$m = \frac{8-0}{0-(-4)} = \frac{8}{4} = 2$

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Question

The grade of a road is defined as the slope of the road expressed as a percent as opposed to a fraction or decimal.

A road is graded so that for every 40 feet of horizontal distance, the road rises 6 feet. What is the grade of the road?

Answer

The slope is the ratio of the vertical change (rise) to the horizontal change (run), so the slope of the road, as a fraction, is $\frac{6}{40}$ . Multiply this by 100% to get its equivalent percent:

$\frac{6}{40} \times 100 % = 15 %$

This is the correct choice.

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Question

What is the slope of the line perpendicular to ?

Answer

In order to find the perpendicular of a given slope, you need that given slope! This is easy to compute, given your equation. Just get it into slope-intercept form. Recall that it is

Simplifying your equation, you get:

$y=\frac{-1}{2}x+\frac{93}{6}$

This means that your perpendicular slope (which is opposite and reciprocal) will be .

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Question

What is the equation of a line with a slope perpendicular to the line passing through the points and ?

Answer

First, you should solve for the slope of the line passing through your two points. Recall that the equation for finding the slope between two points is:

$\frac{rise}{run} = \frac{y_2-y_1}{x_2-x1}$

For your data, this is

$\frac{19-4}{7-2}=\frac{15}{5}=3$

Now, recall that perpendicular slopes are opposite and reciprocal. Therefore, the slope of your line will be $\frac{-1}{3}$ . Given that all of your options are in slope-intercept form, this is somewhat easy. Remember that slope-intercept form is:

is your slope. Therefore, you are looking for an equation with $m=\frac{-1}{3}$

The only option that matches this is:

$y=\frac{-1}{3}x+94$

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Question

What is the x-intercept of ?

Answer

Remember, to find the x-intercept, you need to set equal to zero. Therefore, you get:

Simply solving, this is

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Question

Find the slope of the line that has the equation:

Answer

Step 1: Move x and y to opposite sides...

We will subtract 2x from both sides...

Result,

Step 2: Recall the basic equation of a line...

, where the coefficient of y is .

Step 3: Divide every term by to change the coefficient of y to :

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

Step 4: Reduce...

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

Step 5: The slope of a line is the coefficient in front of the x term...

So, the slope is $\frac {2}{3}$

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Question

Find the slope of the following equation:

Answer

In order to find the slope, we will need the equation in slope-intercept form.

Distribute the negative nine through the binomial.

The slope is:

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Question

What is the y-intercept of the following equation?

Answer

The y-intercept is the value of when .

Substitute the value of zero into , and solve for .

Subtract 7 from both sides.

The answer is:

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Question

Find the slope and y-intercept, respectively, given the following equation:

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

Answer

Rewrite the equation in slope-intercept format:

Divide by negative two on both sides. This is also the same as multiplying both sides by negative half.

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

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

Rearrange the terms.

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

The slope is: $\frac{1}{2}$

The y-intercept is: $\frac{1}{3}$

The answer is: $\frac{1}{2}, \frac{1}{3}$

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Question

Find the slope of the following function:

Answer

Simplify the terms of the equation by distribution.

Subtract the terms.

The equation is now in slope-intercept form, where .

The slope is .

The answer is .

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Question

What is the slope of the following equation?

Answer

To determine the slope, we will need the equation in slope-intercept form.

Subtract on both sides.

Divide by 9 on both sides.

$\frac{9y }{9}= \frac{-x-7}{9}$

Split both terms on the right side.

$y= -\frac{1}{9}x-\frac{7}{9}$

The slope is: $-\frac{1}{9}$

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Question

Identify the y-intercept:

Answer

In order to find the y-intercept, we will need to let and solve for .

Subtract from both sides. Do NOT divide by on both sides.

The answer is:

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Question

What is the x-intercept of the following equation?

Answer

In order to find the x-intercept, we will need to let , and solve for .

Add on both sides.

Divide by five on both sides.

$\frac{5x}{5}=\frac{3}{5}$

The answer is: $\frac{3}{5}$

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Question

A line on the coordinate plane has as its equation .

Which of the following is its slope?

Answer

Rewrite the equation in the slope-intercept form , with the slope, as follows:

Subtract from both sides:

Multiply both sides by $\frac{1}{8}$ , distributing on the right:

$\frac{1}{8} \cdot 8y =\frac{1}{8} \cdot \left ( -3x + 28 \right )$

$y =\frac{1}{8} \cdot ( -3x) +\frac{1}{8} \cdot 28$

$y =\frac{1}{8} \cdot \frac{ -3}{1}\cdot x +\frac{1}{8} \cdot \frac{28}{1}$

$y =- \frac{3}{8} x +\frac{28}{8}$

$y =- \frac{3}{8} x +\frac{7}{2}$

In the slope-intercept form, the coefficient of is the slope . This is $- \frac{3}{8}$ .

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Question

What is the slope of the following function?

Answer

To determine the slope, we will need the equation in slope-intercept form.

Divide by six on both sides.

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

Simplify the right side.

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

The answer is: $-\frac{1}{2}$

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Question

Find the slope of the following equation:

Answer

If we look at the following equation

we can see it is written in slope-intercept form. Slope-intercept form is

where m is the slope and b is the y-intercept.

So, given the equation,

we can see -9 is the slope.

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Question

Find the slope of the following equation:

Answer

The given equation is already in slope-intercept form.

The represents the coefficient of the slope.

The answer is:

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