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GED Math : Linear Algebra

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Example Question #239 : Algebra

Which of the following is an equation, in standard form, of the line of the coordinate plane with intercepts (-3, 0 ) and (0, 8 ) ?

Possible Answers:

8 x - 3 y =-24

8 x - 3 y =24

8 x + 3 y =24

8 x + 3 y =-24

Correct answer:

8 x - 3 y =-24

Explanation:

First, find the slope-intercept form of the equation. This is

y = mx+b ,

where is the slope and (0,b) is the -intercept of the line. Since (0, 8 ) is this intercept, b = 8 . Also, the slope of a line with intercepts (a,0 ) and (0,b) is $m = - \frac{b}{a}$ , so, setting a = -3, b = 8 ,

$m = - \frac{8}{-3} = \frac{8}{3}$ .

The slope-intercept form is

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

The standard form of the equation is

Ax+By = C ,

where, by custom, , , and are relatively prime integers, and A > 0 . To accomplish this:

Switch the expressions:

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

Add -y-8 to both sides:

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

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

Multiply both sides by 3 to eliminate the denominator and make the coefficients integers with GCF 1:

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

Distribute on the left:

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

8 x - 3 y =-24

This is the correct equation.

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Example Question #21 : Standard Form

Rewrite the equation

$y = \frac{2}{7}x- \frac{1}{14}$

in standard form so that the coefficients are integers, the coefficient of is positive, and the three integers are relatively prime.

Possible Answers:

4x+ 14y = 1

14x- 4y = 1

14x+4y = 1

4x-14y =1

Correct answer:

4x-14y =1

Explanation:

The standard form of the equation of a line is

ax+ by = c .

To rewrite the equation

$y = \frac{2}{7}x- \frac{1}{14}$

in this form so that has a positive coefficient, first, switch the places of the expressions:

$\frac{2}{7}x- \frac{1}{14} = y$

Get the term on the left and the constant on the right by adding $-y+ \frac{1}{14}$ to both sides:

$\frac{2}{7}x- \frac{1}{14} + \left ( -y+ \frac{1}{14} \right ) = y+ \left ( -y+ \frac{1}{14} \right )$

$\frac{2}{7}x- y = \frac{1}{14}$

To eliminate fractions and ensure that the coefficients are relatively prime, multiply both sides by lowest common denominator 14:

$14 \cdot \left (\frac{2}{7}x- y \right ) = 14 \cdot \frac{1}{14}$

$14 \left (\frac{2}{7}x- y \right ) = \frac{14}{1} \cdot \frac{1}{14}$

$14 \cdot \left (\frac{2}{7}x- y \right ) =1$

Multiply 14 by both expressions in the parentheses:

$14 \cdot \frac{2}{7}x-14 \cdot y =1$

$\frac{14}{1} \cdot \frac{2}{7}x-14y =1$

Cross-canceling:

$\frac{2}{1} \cdot \frac{2}{1}x-14y =1$

4x-14y =1 ,

the correct choice.

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Example Question #71 : Linear Algebra

What is the standard form of the equation of the line that goes through the point (5, 2) and has a slope of ?

Possible Answers:

4x+y=8

4x+y=-8

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

4x-y=18

Correct answer:

4x-y=18

Explanation:

Start by writing out the equation of the line in point-slope form.

y-2=4(x-5)

Simplify this equation.

y-2=4x-20

Now, recall what the standard form of a linear equation looks like:

Ax+By=C , where $A, B, \text{ and }C$ are integers. Traditionally, is positive.

Rearrange the equation found from the point-slope form so that it has the and terms on one side, and a number on the other side.

y-2=4x-20

y-2-4x=-20

-4x+y=-18

Since the term should be positive, multiply the entire equation by .

4x-y=18

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Example Question #1 : Finding Slope And Intercepts

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

Possible Answers:

$\small slope=-\frac{1}{3}; y-intercept=7$

$\small slope=-3; y-intercept=\frac{1}{7}$

$\small slope=-3; y-intercept=-\frac{1}{7}$

$\small slope=-\frac{1}{3}; y-intercept=-7$

Correct answer:

$\small slope=-\frac{1}{3}; y-intercept=7$

Explanation:

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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Example Question #2 : Finding Slope And Intercepts

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

Possible Answers:

$\textup{slope = }-8\textup{, y-intercept = } \frac{2}{3}$

$\textup{slope = }\frac{3}{2}\textup{, y-intercept = }-8$

$\textup{slope = } 8\textup{, y-intercept = } \frac{2}{3}$

$\textup{slope = }\frac{2}{3}\textup{, y-intercept = }-8$

$\textup{slope = }\frac{2}{3}\textup{, y-intercept = } 8$

Correct answer:

$\textup{slope = }\frac{2}{3}\textup{, y-intercept = }-8$

Explanation:

The slope-intercept form of a line is: y=mx+b , 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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Example Question #3 : Finding Slope And Intercepts

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?

Possible Answers:

$20 \%$

$12 \%$

$18 \%$

$15 \%$

Correct answer:

$15 \%$

Explanation:

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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Example Question #4 : Finding Slope And Intercepts

Line

Refer to above red line. What is its slope?

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

Correct answer:

Explanation:

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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Example Question #1 : Finding Slope And Intercepts

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

3x+2y=-6

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Convert the equation into slope-intercept form, which is y=mx+b , where is the slope and is the y-intercept.

3x+2y=-6

3x-3x+2y=-6-3x

2y=-6-3x

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

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

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

b=-3

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Example Question #6 : Finding Slope And Intercepts

What is the slope of the line perpendicular to 3x + 6y=93 ?

Possible Answers:

$\frac{31}{2}$

$\frac{1}{2}$

$\frac{-1}{2}$

Correct answer:

Explanation:

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 y=mx+b

Simplifying your equation, you get:

6y=-3x+93

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

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

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Example Question #1 : Finding Slope And Intercepts

What is the equation of a line with a slope perpendicular to the line passing through the points (2,4) and (7,19) ?

Possible Answers:

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

y=3x-34

y=5x-15

y=9x+23

$y=\frac{-1}{5}x+17$

Correct answer:

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

Explanation:

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:

y=mx+b

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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GED Math : Linear Algebra

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #239 : Algebra

Example Question #21 : Standard Form

Example Question #71 : Linear Algebra

Example Question #1 : Finding Slope And Intercepts

Example Question #2 : Finding Slope And Intercepts

Example Question #3 : Finding Slope And Intercepts

Example Question #4 : Finding Slope And Intercepts

Example Question #1 : Finding Slope And Intercepts

Example Question #6 : Finding Slope And Intercepts

Example Question #1 : Finding Slope And Intercepts

All GED Math Resources

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