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solve for y, and state the slope and y-intercept? The equation is:
3x - 4y = -12
:
Subtract 3x from both sides
-4y = -3x - 12
:
y has to be positive, mult by -1, change all the signs
4y = 3x + 12
:
Divide both sides by 4
y = x +
y = x + 3
This is the "slope/intercept" form therefore:
slope = , y intercept = 3

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Different amounts are invested at 6%, 7%, and 8% yearly interest.
The amount invested at 7% is $300 more than what is invested at 6%,
and the total yearly income from all three investments is $208.
A total of $2900 is invested.
Find the amount invested at each rate.
:
Let x = amt invested at 6$
Let y = amt invested at 7%
Let z = amt invested at 8%
:
Write an equation for each statement:
:
"The amount invested at 7% is $300 more than what is invested at 6%,"
y = x + 300
:
"the total yearly income from all three investments is $208."
.06x + .07y + .08z = 208
get rid of the decimals, mult by 100
6x + 7y + 8z = 20800
Replace y with (x+300)
6x + 7(x+300) + 8z = 20800
6x + 7x + 2100 + 8z = 20800
13x + 8z = 20800 - 2100
13x + 8z = 18700
:
"A total of $2900 is invested."
x + y + z = 2900
Replace y with (x+300)
x + (x+300) + z = 2900
2x + z = 2900 - 300
2x + z = 2600
:
Two equation two unknowns, use elimination here,
mult the above equation by 8, subtract: 13x + 8z = 18700
16x + 8z = 20800
13x + 8z = 18700
------------------subtraction eliminates z, find x
3x = 2100
x =
x = $700 invested at 6%
:
Find y using, y = x + 300
y = 700 + 300
y =$1000 invested at 7%
:
Find z using, x + y + z = 2900
700 + 1000 + z = 2900
z = 2900 - 1700
z = $1200 invested at 8%
:
:
Check the solution by finding the total interest
.06(700) + .07(1000) + .08(1200) =
42 + 70 + 96 = 208; confirms our solutions of x=700, y=100, z=1200

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An office building has 33 floors, each 10 feet high, and elevators that travel at a constant speed.
If it takes 10 seconds for an elevator to go from the first floor to the fifth floor,
how long will it take an elevator to go from the fifth floor to the middle floor?
:
"it takes 10 seconds for an elevator to go from the first floor to the fifth"
therefore
40 ft in 10 sec; that's 4 ft/sec
:
Middle floor is the 17th
17 - 5 = 12 floors
that's
10 * 12 = 120 ft
then
120/4 = 30 seconds

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A red ball and a green ball are simultaneously tossed into the air.
The red ball is given an initial velocity of 96 feet per second, and its height t seconds after it is tossed is -16t^2 + 96t feet.
The green ball is given an initial velocity of 80 feet per second, and its height t seconds after it is tossed is -16t^2 + 80t feet.
a) find a polynomial D(t) that represents the difference in the heights of the two balls.
:
D(t) = (-16t^2 + 96t) - (-16t^2 +80t)
:
Remove brackets
D(t) = -16t^2 + 96t + 16t^2 - 80t
:
combine like terms:
D(t) = 16t
:
A graph illustrates this well

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+
:
We can factor both denominators (the 1st one is the "difference of squares")
+
:
The common denominator is (x-3)(x+3)(x+1)

:
Multiply inside the brackets

:
Combine like terms in the numerator
=

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AT A CERTAIN FACTORY, PRODUCTS ARE MADE IN VATS THAT HAVE THE CAPACITY TO HOLD 50 QUARTS.
IF EACH BOTTLE OF A PRODUCT CONTAINS 10/26 QUART, HOW MANY BOTTLES CAN BE MADE FROM EACH VAT?
:
We can reduce the fraction to 5/13
:
Let n = no. of qts in the vat
:
n =
invert the dividing fraction and multiply
n = 50 *
Cancel 5
n = 10 * 13 = 130 qts

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There are 112 male and 78 females working at a hospital.
What percentage of the hospitals workforce is female?
:
% female = * 100 ~ 41%

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a 1500 liter tank contains 30 liters of a 70 % acid solution.
You add x liters of a 30 % acid solution to the tank.
Determine a function that represents the proportion of acid to the total
solution in the final mixture
:
y = decimal proportion of acid to the total
y(x+30) = .3x + .7(30)
y(x+30) = .3x + 21
y =

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Trains A and B are traveling the same direction on parallel tracks.
Train A is traveling at 60 miles per hour and train B is traveling at 70 miles per hour.
Train A passes a station at 1:10 P.M.
If train B passes the station at 1:22 P.M. at what time will train B catch up with train A?
:
from the given information, we know that train B is 12 min (1/5 hr) behind
train A, when train A passes the station
:
The distance between the trains at this time: * 70 = 14 mi
:
Let t = time required for train B to catch train A
:
write a distance equation: Dist = speed * time
:
Train B travel dist = Train A travel dist + 14 mi
70t = 60t + 14
70t - 60t = 14
10t = 14
t =
t = 1.4 hr or 1 hr 24 min
:
1:10 + 1:24 = 2:34 pm, B catches A

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You have just turned 12 years old. Your mother promises to give you a car worth $15,000 for your 21st birthday.
You like putting money in your savings account (which earns 3% interest compounded annually),
and tell your mother you would prefer that she just give you some money now to put in the account.
How much should she give you now so that it will grow to $15,000 on your 21st birthday?
:
the compound interest formula for annual compounding,
p*(1+r)^t = A
where
A = resulting amt
p = initial amt (principal)
r = interest rate in decimals
t = time in yrs
:
t = 9 yrs, A = 15000, r = .03; find p
:
p(1.03)^9 = 15000
find 1.03^9 with a calc:
1.304773p = 15000
p =
p = $11,496.25 invested now at 3%

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jason is y years old. his sister jessica is twice as old as jason is.
Their added ages total 30
:
y + 2y = 30
3y = 30
y = 10

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Bird Watchers Club were going to pay $18 each to go on an outing.
Shortly before leaving, 5 new members joined.
With everybody going, this reduced everybody's expense by $2.25.
How many members are now.
:
Let x = no. of members now
then
(x-5) = original no. of members
:
Final amt: 18 - 2.25 = $15.75
:
total cost remains the same therefore:
18(x-5) = 15.75x
:
18x - 90 = 15.75x
:
18x - 15.75x = 90
:
2.25x = 90
:
x = 40 members now
:
:
Check solution by finding the actual total cost
40 * 15.75 = 630
35 * 18 = 630, confirms our solution of x=40

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Patricia lives in a house on a hill. The grade of her street is 9%.
The street ends at a river bank 360 yards from her front door.
Hector, Patricia’s boyfriend, lives in a house 240 yards down from hers.
How much does the river level have to rise about the river bank to flood Hector’s basement?
If Hector’s house is 20 feet tall and the flood covers it completely, would
Patricia’s house suffer any flooding?
:
With a slope that small, the horizontal dist is almost the same as the actual
distance to the river. .09 * 120 = 10.8 yds is H's house above the river
and
.09 * 360 = 32.4 yds is P's house above the river, so even if you add 7 yds
for H's house, P's house is high and dry
:
However if they want more exact value, we have to solve a couple right triangles
:
H's house is: 360-240 = 120 yds from the river
A right triangle formed by:
the horizontal dist to river
the height above the river
the distance from the river we know is 120 yds (hypotenuse)
:
9% grade, the slope rises 9 yd for every 100 yds (horizontally)
Call the horizontal distance x
then
height above the river = .09x
:
Find x using pythag:
x^2 + (.09x)^2 = 120^2
x^2 + .0081x^2 = 14400
1.0081x^2 = 14400
x^2 =
x^2 = 14284.3
x =
x = 119.5 yds is the horizontal distance
then
.09 * 119.5 = 10.75 yds is the height of H's house above the river
The river would have to rise 10.75 yds to flood H's basement
:
"If Hector’s house is 20 feet tall and the flood covers it completely, would
Patricia’s house suffer any flooding?
Convert 20 ft to 6.67 yds, therefore:
10.75 + 6.67 = 17.42 yd rise in the river would cover H's house.
:
How about P's house? Find her height above the river, y = horizontal dist
Same method
y^2 + (.09y)^2 = 360^2
y^2 + .0081y^2 = 129600
1.0081y^2 = 129600
y^2 = 128558.7
y = 358.55 yd is the horizontal dist
then
.09 * 358.55 = 32.27 yd above river, obviously her house is safe

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Build each rational expression into an equivalent rational expression with the indicated denominator.
= ?/4x^2 - 12x + 9
:
Factor 4x^2 - 12x + 9 and we have
(2x-3)(2x-3)
:
If we multiply the 1st denominator by (2x-3), we have the 2nd denominator
:
Multiply the 1st fraction by (the same as mult by 1)
:
* = =

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if my son worked on the garden by himself it would take him 4 hours longer to weed the garden than his grandfather took. However, if he worked with my father-in-law, he worked twice as fast. Develop a formula for how long it took for both of them to weed the whole garden working together. Simplify/reduce to lowest terms.
:
Let x = time required by son working alone
then
(x-4) = time required by grandad alone
:
If son works twice as fast:
.5x = time when son working with grandad
;
Let completed job = 1 (A weeded garden)
:
Let t = time to complete the job working together
:
+ = 1
:
Multiply by .5x(x-4), results
.5xt + t(x-4) = .5x(x-4)
:
.5xt + xt - 4t = .5x^2 - 2x
:
1.5xt - 4t = .5x^2 - 2x
Factor out t
t(1.5x - 4) = .5x^2 - 2x
:
Divide both sides by (1.5x-4)
t = , is the formula
where:
t = time working together
x = son's time working alone

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Solve each system by substitution

:

:
Get rid of those annoying fractions
:
Mult 1st equation by 30, results:
5x + 3y = 90
3y = 90 - 5x
divide thru by 3
y = 30 - x; use this for substitution
:
Mult 2nd equation by 16, results
5x - 3y = 2(15)
5x - 3y = 30
Substitute for y
5x - 3(30 - x) = 30
5x - 90 + 5x = 30
5x + 5x = 30 + 90
10x = 120
x = 12
:
Find y using eq: 5x - 3y = 30; replace x with 12
5(12) - 3y = 30
60 - 3y = 30
-3y = 30 - 60
-3y = -30
y = +10
:
:
Check solutions in the first original equation

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The Clarke family went sailing on a lake.
Their boat averaged 6 kilometers per hour.
The Rourke family took their outboard runabout for a trip on the lake for the same amount of time.
Their boat averaged 14 kilometers per hour.
The Rourke family traveled 10 kilometers farther than the Clarke family.
How many hours did each family spend on their boat trip?
:
Let t = travel time for both boats
:
Write a distance equations: d = speed * time
:
R's dist - C's dist = 10 km
:
14t - 6t = 10
8t = 10
t =
t = 1.25 hrs on their boat trips
:
:
Check solution by finding the distances
14*1.25 = 17.5 km
6*1.25 = 7.5 km
-------------------
differ = 10 km; confirms our solution of t=1.25 hrs

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The reciprocal gets rid of the neg exponent

^(1/2) is the square root

:

The 6th root of 64 is 2

= 32

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Write an equation for each statement
:
Adam has $1.14 in change made up of pennies, nickels and dimes.
.01p + .05n + .10d = 1.14
get rid of the decimals, mult by 100
1p + 5n + 10d = 114
:
He has 6 more nickels than pennies
n = p + 6
:
6 fewer dimes than pennies.
d = p - 6
:
How many of each coin does he have.
Everything in terms of p
:
Using the 1st equation replace n and d using the 2nd and 3rd equation
1p + 5(p+6) + 10(p-6) = 114
:
p + 5p + 30 + 10p - 60 = 114
:
16p - 30 = 114
:
16p = 114 + 30
:
16p = 144
p =
p = 9, pennies
:
I'll let you find n and d, check solution in original 1st equation

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Ms. Vu can run 2 5/8 miles in 1 1/4 miles hour.
At this rate, how many miles can she run in 4 2/5 hour?
:
All these values will translate to exact decimal values(a lot easier to handle)
:
Ms. Vu can run 2.625 miles in 1.25 hours.
At this rate, how many miles can she run in 4.4 hours?
:
Find her speed; speed = dist/time
s =
s = 2.1 mph
:
Dist = speed * time
d = 2.1 * 4.4
d = 9.24 mi or, since they like fractions so much, 9 = 9 mi
:
:
You can also do this as a ratio equation
=
Cross multiply
1.25d = 4.4 * 2.625
1.25d = 11.55
d =
d = 9.24 mi

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In 1992, the life expectancy of males in a certain country was 66.8 years.
In 1996, it was 70.5 years. Let E represent the life expectancy in year t and
let t represent the number of years since 1992.
:
Find the slope
In 1992; x1=0 and y1=66.8
In 1996; x2=4 and y2=70.5
:
Find the slope (m) using the slope equation: m =
m = = = .925
:
Use the point/slope formula to write the equation; y - y1 = m(x - x1)
y - 66.8 = .925(x - 0)
y = .925x + 66.8
:
therefore:
The Linear function E(t) that fits the data is.
E(t)= .925t + 66.8
:
Use the function to predict the life expectancy of males in 2005
t = 2005 - 1992 = 13
E(13) = .925(13) + 66.8
E(13) = 12.025 + 66.8
E(13) = 78.8

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In 1992, the life expectancy of males in a certain country was 66.8 years.
In 1996, it was 70.5 years.
Let E represent the life expectancy in year t and let t represent the numbers of years since 1992.
:
Find the slope
In 1992; x1=0 and y1=66.8
In 1996; x2=4 and y2=70.5
:
Find the slope (m) using the slope equation: m =
m = = = .925
:
Use the point/slope formula to write the equation; y - y1 = m(x - x1)
y - 66.8 = .925(x - 0)
y = .925x + 66.8
:
therefore:
The Linear function E(t) that fits the data is.
E(t)= .925t + 66.8
:

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At the beginning of a walk, Roberto and Juanita are 7.7 miles apart.
If they leave at the same time and walk in the same directions.
Roberto over take jaunita in 11 hours.
If they walk towards each other they meet in one hour.
What are their speeds?
:
let r = R's speed
let j = J's speed
:
Write two distance equations, d = time * speed
:
Walking the same direction
11(r-j) = 7.7
divide both sides by 11
r - j = .7
:
Walking toward each other
1(r+j) = 7.7
r + j = 7.7
:
Add the two equations
r - j = .7
r + j = 7.7
---------------additional eliminates j, find r
2r = 8.4
r =
r = 4.2 mph is R's walking speed
:
Find J's speed using eq: r + j = 7.7
4.2 + j = 7.7
j = 7.7 - 4.2
j = 3.5 mph is J's walking speed
:
:
Check solutions in the "toward" equation
1 * (3.5 + 4.2) = 7.7

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During the first part of a trip, a canoeist travels 87 miles at a certain speed.
The canoeist travels 9 miles on the second part of the trip at a speed 5 mph slower.
The total time for the trip is 5 hours.
What was the speed on the FIRST part of the trip?
What was the speed on the SECOND part of the trip?
(type an integer or a decimal. Round to the nearest hundredth)
:
Let s = "a certain speed on the first part of the speed"
then
(s-5) = "speed on the second part of the trip"
:
:
Write a time equation: Time = dist/speed
:
1st part time + 2nd part time = 5 hrs
+ = 5
:
Multiply equation by s(s-5), results:
87(s-5) + 9s = 5s(s-5)
:
87s - 435 + 9s = 5s^2 - 25s
:
96s - 435 = 5s^2 - 25s
Arrange as a quadratic equation
5s^2 - 25s - 96s + 435 = 0
:
5s^2 - 121s + 435 = 0
Use the quadratic formula to find s:

in this problem: x=s; a=5; b=-121; c= 435

:

:

Two solutions, but this is the one that makes sense

s =
s = 19.81 mph on the first part of the trip
then
19.81 - 5 = 14.81 mph on the 2nd part
:
:
Check solution
+ =
4.39 + .61 = 5 hrs, confirms our solutions

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The chief of police for small town USA notice that, not including himself,
3/5 of his police force carries firearms.
If he include himself, 5/8 of his force carries firearms.
How many people on his force do not carry firearms?
:
Originally then, 2/5 did not carry guns
Later then, 3/8 did not carry guns
:
Let x = total on the police force not counting the chief
:
No. not carrying guns remains unchanged, so we have:
x = (x+1)
Multiply both sides by 40, results
8(2)x = 5(3(x+1))
16x = 15x + 15
16x - 15x = 15
x = 15 total on the police force
then
*15 = 6 did not carry guns
:
Confirm solution
*(15+1) = 6 also

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The population of a certain geographic region is approximately 135 million and
grows continuously at a relative growth rate of 1.87%.
What will the population be in 6 years? Compute the answer to three significant digits
:
Change the growth rate to a decimal; 1 + .0187 = 1.0187
Let P = population (in millions) in 6 yrs
P = 135*1.0187^6
Using a calc
P = 150.873 million in 6 yrs

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chad drove 3 times as fast as katy pedaled.
In fact, chad drove 90 miles in 2 hrs less than it took katy to pedal 40 miles.
How fast did each of them travel. How long did they travel?
:
Let s = K's speed
then
3s = C's speed
:
Write a time equation: Time = dist/speed
:
K's time - C's time = 2 hrs
- = 2
Reduce fraction
- = 2
= 2
Multiply both sides by 2
10 = 2s
s =
s = 5 mph is K's speed
then
2(5) = 15 mph is C's speed
:
Find the travel times
90/15 = 6 hrs
40/5 = 8 hrs
--------------
diff: 2 hrs, confirms our solutions

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An open box is to be made from a 10-ft by 20-ft rectangular piece of cardboard by cutting a square from each corner.
The area of the bottom of the box is to be 96ft^2.
What is the length of the sides of the squares that are cut from the corners?
:
let x = length of the sides of the removed squares
:
Then the dimensions of the box will be: (10-2x) by (20-2x) by x
:
The area of the bottom is given as 96 sq/ft
therefore:
(10-2x)*(20-2x) = 96
FOIL
200 - 20x - 40x + 4x^2 = 96
Arrange as a quadratic equation
4x^2 - 60x + 200 - 96 = 0
4x^2 - 60x + 104 = 0
Simplify, divide by 4
x^2 - 15x + 26 = 0
Factors to
(x-13)(x-2) = 0
Two solutions
x = 13 ft, does not make sense
and
x = 2 ft, is the length of the side of the removed squares
:
:
Check solution by finding the area:
[10 - 2(2)] * [20 - 2(2)] =
6 * 16 = 96

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Find the product or quotient:
:

-----------------

:
each expression can be factored here

-----------------

:
we can cancel some of this immediately, cancel (a+8) and (a-5)

-----------------

:
Invert the dividing fraction and multiply
*
:
Cancel 7 into 35
* (a+4) = or

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Find the quadratic in standard form that goes through (-4,27), (1,2) (3,34)
:
Using the form, ax^2 + bx + c = y, write an equation for each pair
:
x=-4, y=27
a(-4^2) + b(-4) + c = 27
16a - 4b + c = 27
Do the same with the other two pairs
x=1, y=2
a + b + c = 1
:
x=3, y=34
9a + 3b + c = 34
:
Subtract the 2nd equation from the 1st equation
16a - 4b + c = 27
a + b + c = 2
--------------------eliminates c
15a - 5b = 25
simplify divide by 5
3a - b = 5
:
Subtract the 2nd equation from the 3rd equation
9a + 3b + c = 34
a + b + c = 2
--------------------Eliminates c
8a + 2b = 32
simplify, divide by 2
4a + b = 16
:
Two equation which we can handle easily with elimination again
3a - b = 5
4a + b = 16
---------------adding eliminates b, find a
7a = 21
a = 21/7
a = 3
:
Find b using the equation 4a + b = 16,
4(3) + b = 16
12 + b = 16
b = 16 - 12
b = 4
:
Find c using the equation a + b + c = 2
3 + 4 + c = 2
c = 2 - 7
c = -5
:
The equation: y = 3x^2 + 4x - 5
:
You can prove this to yourself by substituting for x and finding y for each pair

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Subtract and simplify if possible

:
The common denominator; (s+d)(s-d), therefore we have

:
Multiply what's inside the brackets

:
Combining like terms we have

:
FOIL the denominator (the difference of squares)

Or