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x + 2 4 x 1 3 = 2

x = −14

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For the following exercises, solve each rational equation for x . State all x -values that are excluded from the solution set.

2 3 x + 4 = x + 2 x + 4

x −4 ; x = −3

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3 x 2 = 1 x 1 + 7 ( x 1 ) ( x 2 )

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3 x x 1 + 2 = 3 x 1

x 1 ; when we solve this we get x = 1 , which is excluded, therefore NO solution

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5 x + 1 + 1 x 3 = 6 x 2 2 x 3

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1 x = 1 5 + 3 2 x

x 0 ; x = −5 2

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For the following exercises, find the equation of the line using the point-slope formula.

Write all the final equations using the slope-intercept form.

( 0 , 3 ) with a slope of 2 3

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( 1 , 2 ) with a slope of 4 5

y = 4 5 x + 14 5

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x -intercept is 1, and ( −2 , 6 )

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y -intercept is 2, and ( 4 , −1 )

y = 3 4 x + 2

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( −3 , 10 ) and ( 5 , −6 )

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( 1 , 3 )   and   ( 5 , 5 )

y = 1 2 x + 5 2

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parallel to y = 2 x + 5 and passes through the point ( 4 , 3 )

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perpendicular to 3 y = x 4 and passes through the point ( −2 , 1 ) .

y = −3 x 5

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For the following exercises, find the equation of the line using the given information.

( 2 , 0 ) and ( −2 , 5 )

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( 1 , 7 ) and ( 3 , 7 )

y = 7

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The slope is undefined and it passes through the point ( 2 , 3 ) .

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The slope equals zero and it passes through the point ( 1 , −4 ) .

y = −4

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The slope is 3 4 and it passes through the point (1,4) .

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( −1 , 3 ) and ( 4 , −5 )

8 x + 5 y = 7

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Graphical

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

y = 2 x + 7 y = 1 2 x 4

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3 x 2 y = 5 6 y 9 x = 6


Coordinate plane with the x and y axes ranging from negative 10 to 10.  The functions 3 times x minus 2 times y = 5 and 6 times y minus 9 times x = 6 are graphed on the same plot.  The lines do not cross.

Parallel

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y = 3 x + 1 4 y = 3 x + 2

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x = 4 y = −3


Coordinate plane with the x and y axes ranging from negative 10 to 10.  The function y = negative 3 and the line x = 4 are graphed on the same plot.  These lines cross at a 90 degree angle.

Perpendicular

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Numeric

For the following exercises, find the slope of the line that passes through the given points.

( 5 , 4 ) and ( 7 , 9 )

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( −3 , 2 ) and ( 4 , −7 )

m = 9 7

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( −5 , 4 ) and ( 2 , 4 )

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( −1 , −2 ) and ( 3 , 4 )

m = 3 2

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( 3 , −2 ) and ( 3 , −2 )

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For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

( −1 , 3 )   and   ( 5 , 1 ) ( −2 , 3 )   and   ( 0 , 9 )

m 1 = 1 3 , m 2 = 3 ; Perpendicular .

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( 2 , 5 )   and   ( 5 , 9 ) ( −1 , −1 )   and   ( 2 , 3 )

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Technology

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y -intercept occurs. State your ymin and ymax values.

0.537 x 2.19 y = 100

y = 0.245 x 45.662. Answers may vary. y min = −50 ,   y max = −40

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4,500 x 200 y = 9,528

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200 30 y x = 70

y = 2.333 x + 6.667. Answers may vary. y min = −10 ,   y max = 10

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Extensions

Starting with the point-slope formula y y 1 = m ( x x 1 ) , solve this expression for x in terms of x 1 , y , y 1 , and m .

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Starting with the standard form of an equation A x  + B y  = C, solve this expression for y in terms of A , B , C , and x . Then put the expression in slope-intercept form.

y = A B x + C B

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Use the above derived formula to put the following standard equation in slope intercept form: 7 x 5 y = 25.

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Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular.

( −1 , 1 ) , ( 2 , 0 ) , ( 3 , 3 ) , and ( 0 , 4 )

The slope for  ( −1 , 1 )  to  ( 0 , 4 )  is  3. The slope for  ( −1 , 1 )  to  ( 2 , 0 )  is  1 3 . The slope for  ( 2 , 0 )  to  ( 3 , 3 )  is  3. The slope for  ( 0 , 4 )  to  ( 3 , 3 )  is  1 3 .

Yes they are perpendicular.

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Find the slopes of the diagonals in the previous exercise. Are they perpendicular?

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Real-world applications

The slope for a wheelchair ramp for a home has to be 1 12 . If the vertical distance from the ground to the door bottom is 2.5 ft, find the distance the ramp has to extend from the home in order to comply with the needed slope.

30 ft

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If the profit equation for a small business selling x number of item one and y number of item two is p = 3 x + 4 y , find the y value when p = $ 453  and   x = 75.

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For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be y = 45 + .25 x , where x is the number of miles traveled.

What is your cost if you travel 50 mi?

$57.50

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If your cost were $ 63.75 , how many miles were you charged for traveling?

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Suppose you have a maximum of $100 to spend for the car rental. What would be the maximum number of miles you could travel?

220 mi

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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