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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

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Nyibol Reply
what is microbiologist
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what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
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studies of microbes
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How bacteria create energy to survive?
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Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
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Muhamad
they make spores
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the significance of food webs for disease transmission
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food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
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This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
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Examples of thermophilic organisms
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Give Examples of thermophilic organisms
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Prevent foreign microbes to the host
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they provide healthier benefits to their hosts
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They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
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cell is the smallest unit of life
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cell is the structural and functional unit of life
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is the fundamental units of Life
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There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
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I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
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Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
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skin
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skin,Oral,Nasal,GIt
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part of a tissue or an organ being wounded or bruised.
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Binomial nomenclature
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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