You can put this solution on YOUR website!
The average life expectancy for American men born in 1900 was 55 years.
Life expectancy has increased by about 0.2 year for each birth year after
1900.
If this trend continues, for which birth year will the average life expectancy be 91 years?
:
Using the form: y = mx + b
From the given information we can say the slope (m) of the equation will be +.2
(slope is rise/run; it goes up .2 for each year)
:
the y intercept (b) will be 55, so we have
y = .2x + 55
Where
y = life expectancy x years after 1900
:
"If this trend continues, for which birth year will the average life expectancy be 91 years?"
replace y with 91, solve for x
.2x + 55 = 91
.2x = 91 - 55
.2x = 36
x =
x = 180 yrs after 1900 which is 2080, so you thought correctly.

You can put this solution on YOUR website!
John leaves his house at 10:00am and drives east at 50 miles per hour.
Mike leaves his house at the same time and drives west at 45 miles per hour.
If their houses are connected by a straight highway running east-west (no
stoplights or towns in the way) and are 350 miles apart then:
:
(a) how far from John's house will they meet?
Let x = distance from John's house (and distance driven by John)
then
(350-x) = distance driven by Mike
:
When they meet they will have driven a total of 350 mi:
Their travel times will be the same; write a time equation; Time =
John's travel time = Mike's travel time
=
Cross multiply:
45x = 50(350-x)
:
45x = 17500 - 50x
:
45x + 50x = 17500
x =
x = 184.21 mi from John's house, they will meet
:
(b) at what time will they meet?
Find the travel time of each
= 3.68 hrs which is 3 + .68(60) = 3 hr 41 min
10 am + 3hr 41min = 1:41 pm
:
;
We can confirm our solution by finding Mikes travel time, it should equal John's
= 3.68 hrs also

You can put this solution on YOUR website!
An airplane flew for 4.23 hours with a 25.5-km/h tail wind.
The return flight against the same wind took 4.97 hours.
Find the speed of the plane in still air.
:
Let s = speed of the plane in still air
then
(s+25.5) = ground speed with the wind
and
(s-25.5) = ground speed against the wind
:
The distance there and back is equal, write a dist equation; Dist = time * speed
:
4.23(s+25.5) = 4.97(s-25.5)
4.23s + 107.865 = 4.97s - 126.735
107.865 + 126.735 = 4.97s - 4.23s
234.6 = .74s
s =
s = 317.03 km/hr speed in still air
;
;
Check solution by finding the distances:
4.23(317.03+25.5) = 4.97(317.03-25.5)
4.23(342.53) = 4.97(291.53)
1448.9 = 1448.9

You can put this solution on YOUR website!
It takes Terry 2 hours longer to do a certain job than it takes Tom. They worked together for 3 hours; then Tom left and Terry finished the job in 1 hour. How long would it take each of them to do the job alone.
:
let x = time required by Tom
then
(x+2) = time required by Terry
:
Let the completed job = 1
:
Solve this equation (Terry worked a total of 4 hrs):
+ = 1

You can put this solution on YOUR website!
Suppose that Arlene can mow the entire lawn in 40 minutes less time with the
power mower than she can with the push mower. One day the power mower broke
down after she had been mowing for 30 minutes. She finished the lawn with
the push mower in 20 minutes. How long does it take Arlene to mow the entire
lawn with the power mower:
:
Let t = time required to mow the lawn with the power mower (in minutes)
then
(t+40) = time required with the push mower
:
Let the completed job = 1
;
+ = 1
multiply equation by t(t+40) to get rid of the denominators, results:
30(t+40) + 20t = t(t+40)
:
30t + 1200 + 20t = t^2 + 40t
:
50t + 1200 = t^2 + 40t
:
0 = t^2 + 40t - 50t - 1200
A quadratic equation:
t^2 - 10t - 1200 = 0
Factors to:
(t - 40)(t + 30) = 0
Positive solution is what we want here:
t = 40 minutes to mow the lawn with power
;
:
Check solution:
+ =
+ = 1

You can put this solution on YOUR website!
Mr. Carlson took 2.5 hours to fly his airplane a distance of 360 miles against a steady wind.
It took him 2 hours to make the return trip with the wind.
Find Mr. Carlson's rate in calm air, and the rate of the wind. use r=d/t.
:
Let r = his speed in calm air
Let w = the speed of the wind
then
(r+w) = his ground speed with the wind
and
(r-w) = his ground speed against the wind
:
Write a distance equation for each trip: Dist = time * speed
2.5(r-w) = 360
2.0(r+w) = 360
Simplify both equations, divide the 1st by 2.5 and the second by 2, results:
r - w = 144
r + w = 180
-------------Addition eliminates w, find r
2r = 324
r =
r = 162 mph in still air
:
and using r + w = 180, find w
162 + w = 180
w = 180 - 162
w = 18 mph is the wind
;
:
Check solutions in the 1st original equation
2.5(162 - 18) =
2.5(144) = 360 mi, confirms our solutions

You can put this solution on YOUR website!
David bought 3 hamburgers and 2 servings of french fries at Diane's Drive -Up for $6.25.
If a serving of fries costs .25 cents more then a hamburger, what is the cost of each?
:
Let h = cost of a hamburger
Let f = cost of an order of fries
:
Write an equation for each statement
"David bought 3 hamburgers and 2 servings of french fries at Diane's Drive -Up for $6.25."
3h + 2f = 6.25
:
"If a serving of fries costs .25 cents more then a hamburger,"
f = h + .25
:
what is the cost of each?
:
we know; 3h + 2f = 6.25
Substitute (h+.25) for f in this equation:
3h + 2(h+.25) = 6.25
3h + 2h + .50 = 6.25
5h = 6.25 - .50
5h = 5.75
h =
h = $1.15 for a hamburger
:
Find f using the 1st equation; f = h + .25
f = 1.15 + .25
f = $1.40 for fries
;
:
you can prove it to yourself: 3h + 2f = 6.25, therefore:
3(1.15) + 2(1.40) =
3.45 + 2.80 = 6.25

You can put this solution on YOUR website!
the square root of x minus 2 minus 7 equals negative 4
Assume the problem is:
= -4
:
Add 7 to both sides:
= -4 + 7
= 3
Square both sides;
x - 2 = 9
x = 9 + 2
x = 11
:
:
Check solution in original equation
= -4
= -4

You can put this solution on YOUR website!
A man walk a certain distance at certain rate. Had he walked 1/2 km an hour
faster he would have taken 1 hour less, but he had gone 1 km an hour slower,
he would have taken 3 hour longer. find the distance he traveled and his original rate.
:
let d = distance traveled
:
let r = original rate of travel
and
d/r = actual time
:
then
(r-1) = the slower rate
and
(r+.5) = the faster rate
:
Write Two time equations; Time =
Eq1; slower time - actual time = 3 hr
- = 3
multiply equation by r(r-1) to get rid of the denominators, results:
dr - d(r-1) = 3r(r-1)
dr - dr + d = 3r^2 - 3r
d = 3r^2 - 3r
:
eq2; actual time - fast time = 1 hr
- = 1
Multiply equation by r(r+.5) to get rid of the denominator, results
d(r+.5) - dr = r(r+.5)
dr + .5d - dr = r^2 + .5r
.5d = r^2 + .5r
Multiply equation by 2
d = 2r^2 + r
:
d = d, therefore we can say:
3r^2 - 3r = 2r^2 + r
Divide equation by r
3r - 3 = 2r + 1
3r - 2r = 1 + 3
r = 4 km/h actual waking speed
:
Find the distance traveled:
d = 2r^2 + r
Substitute 4 for r
d = 2(4^2) + 4
d = 2(16) + f
d = 36 km
;
;
Check distance in the 2nd equation
d = 3r^2 - 3r
Substitute 4 for r
d = 3(4^2) - 3(4)
d = 48 - 12
d = 36 km
:
:
Confirm times:
Actual time = 36/4 = 9 hrs
Faster time = 36/4.5 = 8 hrs (1 hr faster)
slower time = 36/3 = 12 hrs (3 hrs slower)
:
:
Hope all this made sense to you.

You can put this solution on YOUR website!
Debbie traveled by boat 5 miles upstream to fish in her favorite spot.
Because of the 4 mph current, it took her 20 minutes longer to get there
than to return.
How fast will her boat go in still water?
:
Let s = boat speed in still water
then
(s-4) = speed upstream
and
(s+4) = speed down stream
:
Convert 20 min to hrs: 20/60 = hr
:
Write a time equation: time =

Upstream time = downstream time + 20 minutes (1/3 hr)
= +
:
To clear out these denominators, multiply equation by 3(s+4)(s-4)
3(s+4)(s-4)* = 3(s+4)(s-4)* + *3(s+4)(s-4)
Results in
:
15(s+4) = 15(s-4) + (s+4)(s-4)
:
15s + 60 = 15s - 60 + s^2 - 16
:
0 = 15s - 15s - 60 - 60 + s^2 - 16
:
0 = s^2 - 136
:
s^2 = 136
s =
s = 11.66 mph speed in still water
:
:
Check solution on calc: (speed up = 7.66 and speed down = 15.66)
5/7.66 - 5/15.66 = .3333 which is 1/3 of an hr

You can put this solution on YOUR website!

:
Combine the exponents of like terms in a manner that will ensure positive exponents


;
When you raise one exponent to another exponent, multiply

You can put this solution on YOUR website!
y = 1/2x - 2/3
:
Get rid of the fractions, multiply both sides by 6
6y = 3x - 4
:
Subtract 3x from both sides, results
-3x + 6y = -4
Multiply equation by -1 to change the signs
3x - 6y = 4

You can put this solution on YOUR website!
Devon purchased tickets to an air show for 9 adults and 2 children.
The total cost was 192 dollars.
The cost of a child's ticket was 3 dollars less than the cost of an adult's ticket.
Find the price of an adult's ticket and a child's ticket
:
Let x = cost of an adult ticket
then
(x-3) = cost of a child's ticket
;
9x + 2(x-3) = 192
:
9x + 2x - 6 = 192
11x = 192 + 6
11x = 198
x =
x = $18 for an adult ticket
then
18 - 3 = $15 for a child's ticket
;
;
Check:
9(18) + 2(15) = 192

You can put this solution on YOUR website!
The costs of doing business for a company can be found by adding fixed costs, such as rent, insurance, and wages, and variable costs, which are the costs to purchase the product you are selling. The portion of the company’s fixed costs allotted to this product is $300, and the supplier’s cost for a set of tile is $6 each. Let x represent the number of tile sets.
:
If b represents a fixed cost, what value would represent b? "You can see from
the previous statement, that fixed cost is given as $300, therefore b = 300"
:
Find the cost equation for the tile. Write your answer in the form C = mx + b.
"The cost would be the cost for each tile, given as $6, times the number of
tiles which is x, plus the fixed cost, given as b," hence:
C = 6x + 30
:
The profit made from the sale of tiles is found by subtracting the costs from the revenue. " write this as Profit = R - C, substitute our cost equation for C
P = R - (6x + 30)
:
Find the Profit Equation by substituting your equations for R and C in the equation . Simplify the equation.
:
We don't know the revenue unless they tell us what the retail sale amt is for each tile.
For example if they sell each tile for $12. Revenue = 12x
then we can say: profit:
P = 12x - (6x + 300)
Simplify:
p = 12x - 6x - 300
p = 6x - 300
:
:
Did this help?

You can put this solution on YOUR website!
Let x = "one number"
Let y = "a second number"
:
Write an equation for each statement:
:
"one number is four more than a second number."
x = y + 4
:
" two times the first number is four more than four times the second number"
2x = 4y + 4
;
From the 1st equation, substitute (y+4) for x in the above equation
2(y + 4) = 4y + 4
2y + 8 = 4y + 4
8 - 4 = 4y - 2y
4 = 2y
y =
y = 2, the second number
and
x = 2 + 4
x = 6, a number
:
:
Check solution in 2nd statement:
"two times the first number is four more than four times the second number"
2(6) = 4(2) = 4

You can put this solution on YOUR website!
The credit remaining on a phone card (in dollars) is a linear function of the
total calling time made with the card (in minutes), as shown in the figure below.
The remaining credit after 45 minutes of calls is $22.80 , and the remaining
credit after 64 minutes of calls is $19.76.
What is the remaining credit after 73 minutes of calls?
;
Not sure what figure you are referring to, write an equation from the given values:
Assign the values as follows
x1 = 45; y1 = 22.80
x2 = 64; y2 = 19.76
:
Find the slope (m) using the formula: m =
m = =
m = -.16
:
Write the equation using the point/slope formula: y - y1 = m(x - x1)
y - 22.80 = -.16(x - 45)
y - 22.80 = -.16x + 7.2
y = -.16x + 7.2 + 22.80
y = -.16x + 30
;
Use this equation to find the credit (y) when minutes (x) = 73
y = -.16(73) + 30
y = -11.68 + 30
y = $18.32; credit after 73 minute of calls
:
:
That you can also see the original value of the phone card: x=0, y = $30
;
:
Did this help you understand what's going on here? Any questions

You can put this solution on YOUR website!
x =
multiply both sides by (x-3), results:
x(x-3) = 10
:
x^2 - 3x = 10
:
x^2 - 3x - 10 = 0
Factors to:
(x-5)(x+2) = 0
Two solutions:
x = +5
x = -2
;
:
Check both in original equation:
5 =
and
-2 =

You can put this solution on YOUR website!
=
Cross multiply
2(5t-12) = t^2
:
10t - 24 = t^2
:
0 = t^2 - 10t + 24
A quadratic equation that can be factored to:
(t-4)(t-6) = 0
two solutions:
t = +4
and
t = +6
:
:
Check both solutions in original problem
x=6
=
=
=
You can check solution x=4 in the same way

You can put this solution on YOUR website!
One third of a certain number exceeds 12 by as much as 100 exceeds the number. What is the number?
:
Let x = " a certain number"
x = 12 + (100-x)
;
x = 112 - x
Multiply equation by 3
x = 336 - 3x
:
x + 3x = 336
:
4x = 336
x =
x = 84
:
:
Is that true?
*84 = 12 + 100 - 84
28 = 28

You can put this solution on YOUR website!
Owners of a recreation area are filling a small pond with water. They are adding water at a rate of 35 liters per minute. There are 700 liters in the pond to start.
Let W represent the amount of water in the pond (in liters), and let T represent the number of minutes that water has been added. Write an equation relating W to T, and then graph your equation using the axes below.
:
the equation would be: Total water (W) = 35L*minutes(T) + 700L
W = 35T + 700
:
Put T on the horizontal (x) and W on the vertical (Y)
Graph should look like this:

Note when T = 0, you have 700L, when T = 60, you have 35(60)+ 700 = 2800L

You can put this solution on YOUR website!
solve these simultaneous equations:
5x - 5y = 5
4x - 3y = 25
:
Take the 1st equation:
5x - 5y = 5
Simplify, divide by 5:
x - y = 1
x = y + 1; use this for substitution for x in the 2nd equation:
:
4x - 3y = 25
4(y+1) - 3y = 25
4y + 4 - 3y = 25
4y - 3y = 25 - 4
y = 21
:
Find x:
x = y + 1
x = 21 + 1
x = 22
:
:
Check solution in the 2nd equation
4(22) - 3(21) = 25
88 - 63 = 25; confirms our solutions

You can put this solution on YOUR website!
Ramon can do the weekly billing in 2 hrs; Luann needs three hours. If Romon
works for 30 minutes, and then is joined by Luann, how long will it take them
to finish?
:
Let t = amt of time required for them to finish
:
Change 30 min to .5 hrs
:
Let the completed job = 1
:
+ = 1
Multiply equation by 6 to get rid of denominators, results:
3(t+.5) + 2t = 6
:
3t + 1.5 + 2t = 6
:
5t = 6 - 1.5
:
5t = 4.5
t =
t = .9 hrs which is .9(60) = 54 min to finish the job
;
Check solution:
+ =
+ =
.7 + .3 = 1
:
:
the tens digit of a two digit number is one more than 3 times the unit digit.
:
Let x = the 10's digit, and let y = units
then
10x + y = the two digit number
:
Write an equation for the statement:
"the tens digit of a two digit number is one more than 3 times the unit digit."
x = 3y + 1
:
subtracting 45 from the number reverses it ,
10x + y - 45 = 10y + x
10x - x = 10y - y + 45
9x = 9y + 45
Simplify, divide equation by 9
x = y + 5
:
find the original number
:
Substitute (3y+1) for x in the above equation
3y + 1 = y + 5
3y - y = 5 - 1
2y = 4
y = 2
then
x = 2 + 5
x = 7
:
72 is our number
;
;
Check solution in the statement:"subtracting 45 from the number reverses it"
72 - 45 = 27

You can put this solution on YOUR website!
Some US one cent coins are made mostly of copper (each has a mass of 3.1 grams).
Some US one cent coins are made mostly of zinc (each has a mass of 2.5 grams).
Mr. Kniser has a bag of 1000 US one cent coins. The total mass of the bag is 2616.4 grams.
How many of each type of cent is in the bag?
:
Let c = no. of copper coins
Let z = no. of zinc coin
:
A system of two equations:
:
total no. of coins
c + z = 1000
which can be written:
c = (1000-z)
:
and total weight
3.1c + 2.5z = 2616.4
:
Substitute (1000-z) for c in the above equation:
3.1(1000-z) + 2.5z = 2616.4
3100 - 3.1z + 2.5z = 2616.4
-.6z = 2616.4 - 3100
-.6z = -483.6
z =
z = +806 zinc coins
and
1000 - 806 = 194 copper coins
;
:
Check solution in the weight equation
3.1(194) + 2.5(806) =
601.4 + 2015 = 2616.4

You can put this solution on YOUR website!
An adjustable water sprinkler that sprays water in a circular pattern is placed at the center of a square field whose area is 1250 square feet. What is the shortest radius setting that can be used if all of the grass is to be watered?
:
We want to find the diagonal (d) of the square, that would be the diameter
of the circular watered area. The radius would be half of that.
:
Since it's a square, the side dimensions would be:
s =
then we square that to find the hypotenuse: d^2 = s^2 + s^2, so we have:
d^2 = 1250 + 1250
d =
d = 50 ft
then
r = 25 ft is the minimum radius of the sprinkler

You can put this solution on YOUR website!
In a test of a heat-seeking rocket, the target rocket is launched at 2000 ft/sec
and the heat-seeking rocket is launched along the same flight path 12 seconds
later at a speed of 3200 ft/sec. Find the times t1 and t2 of flight of the
rockets until the heat-seeking rocket destroys the first rocket.
:
From the given information we can say:
t1 = t2 + 12 (seconds)
:
When destruction occurs, both rockets will have traveled the same distance
write a distance equation from this fact: Dist = speed * time
:
2000(t1) = 3200(t2)
t1 =
t1 = 1.6(t2)
Substitute (t2-12) for t1; find t2
t2 + 12 = 1.6t2
12 = 1.6t2 - t2
12 = .6t2
t2 =
t2 = 20 seconds
then
t1 = 20 + 12 = 32 seconds
;
:
Check solution by ensuring they both traveled the same distance
2000 * 32 = 64000 ft
3200 * 20 = 64000 ft also

You can put this solution on YOUR website!
An electronics company has fixed costs of $25,000 per month and a variable cost
of $18.65 per 13 inch tv/vcr combination manufactured.
(Fixed costs are those that occur regardless of the level of production)
:
a) Write the total monthly costs C as a function of the number of units x produced.
C = 18.65x + 25000
:
:
b) Use a graphing utility to graph the cost function.
Enter y = 18.65x + 25000, Scale: x:-2000,+15000. y:-50000, +300000
Should look like this:

c) Use the graph from part b to approximate the number of units that can be produced per month if total costs cannot exceed $200,000.
:
Green line represents 200000
:
Verify algebraically.
18.65x + 25000 =< 200000
18.65x =< 200000 - 25000
18.65x =< 175000
x =
x = 9383.37 ~ 9,383 units
;
Is this problem better solved algebraically or graphically? Explain.
:
I'll let you do that.

You can put this solution on YOUR website!
You have submitted quite a few problems here. I will do the first one and give
you the equations for the rest. You can email me if you have difficulty.
:
1. An investor has three investments which pay 3%, 5% and 8%, respectively.
He has three times as much invested at 5% than at 3% and$ 600 more invested
at 8% than at 3%, if the total interest from the investment after 1 year is
$854, how much was invested at 5%?
:
Let x = amt invested at 3%
then
3x = amt invested at 5%
and
(x+600) - amt invested at 8%
;
Write a total interest equation:
.03x + .05(3x) + .08(x+600 ) = 854
.03x + .15x + .08x + 48 = 854
.26x = 854 - 48
x =
x = $3100 invested at 3%
and
2(3100) = $9300 invested at 5%
:
:
2. A solution problem: How many ounces of a 25% solution must be added to 40 ounces of a 7% saline solution to make a 15% saline solution?
:
Let x = amt of 25% solution required
then
(x+40) = resulting amt:
:
.07(40) + .25(x) = .15(x+40)
:
:
3. How many cups of party mix that is 69% pretzels must be added to 120 cups of a party mix that is 45% pretzels to make a party mix that is 54% pretzels?
:
let x = 69% mix
then
(x+120) = resulting amt
:
.69x + .45(120) = .54(x+120)
:
:
4. A merchant has coffee worth $30 a pound that she wishes to mix with 90 pounds of coffee worth $90 a pound to get a mixture that can be sold for $70 a pound how many pounds of $30 coffee should be used?
:
Let x = amt of $30 coffee required
then
(x+90) = resulting amt
:
30x + 90(90) = 70(x+90)
;
:
5. If three times the smaller of two consecutive integers is added to four times the larger, the result is 116. Find the smaller integer.
:
Two consecutive integers: x, (x+1)
:
3x + 4(x+1) = 116
;
The answer is indeed, 16, way to go, substitute 16 for x in the equation to
prove it to yourself

You can put this solution on YOUR website!
The demand function for an office supply company's line of plastic rulers is
p = 0.45 - 0.00045q, where p is the price (in dollars) per unit when q units
are demanded(per day) by consumers.
I think it should be:
Find the level of production that will maximize the manufacturer's total
revenue, and determine this revenue.
:
p = 0.45 - 0.00045q
:
Level of production = q
:
Revenue = quantity * price
r = q * p
;
Replace p with (.45-.00045q) in the above equation and you have
r = q * (.45-.00045q)
r = .45q - .00045q^2
Arrange as a quadratic equation:
r = -.00045q^2 + .45q
:
We can find the value of q which gives max amt by using the eq: x = -b/(2a)
in this equation that would be
q =
q =
q = +500 units need to be produced for max revenue
:
Then it says,"and determine this revenue"
Substitute 500 for q in the revenue equation (r = -.00045q^2 + .45q)
r = -.00045(500^2) + .45(500)
r = -.00045(250000) + 225
r = -112.5 + 225
r = $112.5 is the max revenue and occurs when you produce 500 units
:
Did this make sense to you, any questions?

You can put this solution on YOUR website!
In a basketball game, the bulldogs scored 34 field goals, each counting 2 points, and 13 foul goals, each counting 1 point. The panthers scored 38 field goals and 8 foul goals, which team won? By how many points did it win?
:
Panthers: 2(38) + 1(8) = 84
Bulldogs: 2(34) + 1(13) = 81
------------------------------
difference between them = 3 points
:
I have no idea what the "Polya 4 step method" is

You can put this solution on YOUR website!
In a sports league of n teams in which each team plays every other team twice, the total number N of games to be played by: N = n^2 – n.
:
How many teams are in a softball league if the total number of games played is 42?
:
n^2 - n = 42
:
n^2 - n - 42 = 0; A quadratic equation
factors to:
(n-7)(n+6) = 0
The positive solution
n = 7 teams
:
:
Check solution in original equation:
N = 7^2 - 7
N = 42

You can put this solution on YOUR website!
: Suppose z varies directly as x and inversely as the square of y.
z =
:
When x = 35 and y = 7, the value of z is 50.
Find k using the above values
= z
= 50
=
=
k = * 49
k = 10 * 7
k = 70
Choose the function that models the relationship and z when x = 5 and y = 10
:
z = is the function
z =
z =
z = 3.5