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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

the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
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Practice Key Terms 7

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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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