Questions on Word Problems: Linear Equations And Systems Word Problems answered by real tutors!

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C=0.000015x^2 -0.03x+35
.
Since, the first coefficient (0.000015) is POSITIVE the equation is a parabola that opens upwards. This then says that finding the "vertex" will give you the minimum.
.
The vertex is then found by:
x = -b/2a
x = -(-0.03)/2(0.000015)
x = 0.03/2(0.000015)
x = 0.03/0.00003
x = 1000 (minimum number of tires)
.
The cost, plug it back into:
C=0.000015x^2 -0.03x+35
C=0.000015(1000)^2 -0.03(1000)+35
C=15 -30+35
C=15 +5
C= $20

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OK, first take a deep breath.
Let's get started.
.
.
.
Name your variables.
G-weight of a goblet
C-weight of a cup
P-weight of a plate
.
.
.
Next, write down what you know, in mathematical terms.
.
.
.
"Two plates and a cup together weigh more than three goblets"
1.
.
.
.
"Two goblets and a cup weigh more than two plates"
2.
.
.
.
From 1,
1.

Let's substitute that into equation 2,
2.
Now add what we learned above,

and remove the 2P from the middle,

Add C to both sides and subtract 2G fom both sides,

or

.
.
.
A goblet weighs less than two cups.

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Total square inches of the area would be
:
7 (5.5)=38.5 sq in
:
but we must subtract out the middle portion of 1.2 inches by .3 inches in order to get the light switch area.
:
1.2 (.3)=.36sq in
:
38.5-.36=38.14 sq in. at a cost of $4.27 a sq in would be
:
38.14 (4.27)=$ 162.29

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Let J = Jane's age
and B = Beth's age
.
From: "3 time's Jane's age, in years, is equal to 8 times Beth's age"
3J = 8B (equation 1)
.
From: "difference between their ages is 15 years"
J-B = 15 (equation 2)
.
Solving equation 2 for J:
J-B = 15
J = 15+B
.
Plug the above into equation 1 and solve for B:
3J = 8B
3(15+B) = 8B
45+3B = 8B
45 = 5B
9 = B (Beth's Age)
.
Plug the above into equation 2 and solve for J:
J-B = 15
J-9 = 15
J = 24 (Jane's Age)

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(3,9),(5,13),(15,33),(34,71),(678,1359), and (1234,2471) all lie on line M.
First we find the slope=(y2-y1)/(x2-x1)
Slope=(13-9)/(5-3)=4/2=2 for this slope.
Now replace one set of (x,y) values in the line equation Y=mX+b & solve for the y intercept (b).
9=2*3+b
9=6+b
b=9-6
b=3 the y intercept for line M.
Thus the equation for this line M is:
y=2x+3 Equaton for line M (red line)
--------------------------------------
(3,-9),(5,-11),(15,-21),(34,-40),(678,-684), and (1234,-1240) all lie on line N.
Doing the for these points & line:
slope=(-11+9)/(5-3)=-2/2=-1 for this slope.
-9=-1*3+b
-9=-3+b
b=-9+3
b=-6 the y intercept.
Thus the equation for this line is:
y=-x-6 Equation for line N (green line)
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, of TWO functions 2x +3 and -x -6).
Intercept=(-3,-3)
X axis intercept for M=-1.5, for N=-6
Y axis intercept for M=3, for N=-6

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.
Let x = time (in hours) it takes all three to eat one sheep
.
x/4 + x/5 + x/6 = 1
30x + 24x + 20x = 120
74x = 120
x = 120/74
x = 1.622 hours
or
1 hour 37 minutes and 18 seconds

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We have the following ratio:

       Cost                   Cost
------------------  = --------------------
  Number of Hours        Number of Hours

So in this case, the ratio is:




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



Start with the given ratio


Multiply both sides by 39


Multiply


Reduce



So our answer is which means that the lawyer will charge $7,800 for 39 hours of service.

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Ken mails three packages at the post office that weigh 8.1, 8.5, and 8.9 lbs. The post office that Ken uses rounds the weight of a package to the nearest pound to determine the price of shipment It costs $2 to mail a package that weighs 8 pounds and $2.75 for packages that weigh 9 lbs.
---------

a. How much did Ken pay to ship each package?
8.1 lbs. rounds to 8 lbs which costs $2
8.5 lbs. rounds to 8 lbs which costs 2
8.9 lbs. rounds to 9 lbs which costs 2.75
-------------------------------------------------------
b. Ken has $7.25 in his wallet in cash. Use subtraction to determine whether Ken has enough to mail the three packages and how much extra money he has or needs.
Extra money = 7.25 - (2+2+2.75) = 7.25 - (6.75) = 50 cents
====================
Cheers,
Stan H.

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1st condition: By Ratio:

2nd Condition:
800,000(Gn)+1,100,000(Bl)=18,600,000
Via 1st Condtn we get:
---> 3rd Condition, susbt. in 2nd Contn.:
.
800,000(5/2)Bl+1,100,000(Bl)=18,600,000
2,000,000(Bl)+1,100,000(Bl)=18,600,000
3,100,000(Bl)=18,600,000, Isolate "Black" by dividing both terms 3,100,0000

---> number of Black jeeps
Via condition 3:
---> number of Green Jeeps
In doubt? Go back Condtn. 2 (amount of fuel consumed):


, good
Thank you,
Jojo

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here's how i solved it:
let s = stocks
let b = bonds
let m = money market funds
-----
s + b + m = 45000
.10*s + .07*b + .075*m = 3660
-----
"The total she invested in bonds and money market funds was twice the amount she invested in stocks"
2s = (b+m)
divide both sides of equation by 2:
s = (b+m)/2
-----
i started off with 2 equations in 3 unknowns.
using the fact of s = (b+m)/2, i was able to substitute (b+m)/2 for s which created 2 equations in 2 unknowns.
-----
substituting (b+m)/2 for s in both equations gets:
(b+m)/2 + b + m = 45000
.10*(b+m)/2 + .07*b + .075*m = 3660
-----
multiplying both equations by 2 gets:
(b+m) + 2b + 2m = 90000
.10*(b+m) + .14*b + .15*m = 7320
-----
remove parentheses from both equations:
b + m + 2b + 2m = 90000
.10*b + .10*m + .14*b + .15*m = 7320
combine like terms:
3b + 3m = 90000
.24*b + .25*m = 7320
-----
multiply the second equation by 12. this makes 3m in both equations so the m can cancel out and we can solve for b.
-----
3b + 3m = 90000
2.88b + 3m = 87840
-----
subtract the second equation from the first equation:
.12b = 2160
divide boths sides of the equation by .12
-----
b = 2160/.12 = 18000
-----
substitute 18000 for b in the first equation after you combined like terms.
3b + 3m = 90000
3*18000 + 3m = 90000
54000 + 3m = 90000
subtract 54000 from both sides:
3m = 90000 - 54000 = 36000
divide both sides by 3:
m = 36000 / 3 = 12000
-----
you have so far:
b = 18000
m = 12000
solve for s by substituting in the original equation shown below:
s + b + m = 45000
s + 18000 + 12000 = 45000
combine like terms:
s + 30000 = 45000
subtract 30000 from both sides:
s = 45000 - 30000 = 15000
-----
you now have all three:
s = 15000
b = 18000
m = 12000
-----
substitute in the second original equation as shown below:
.10*s + .07*b + .075*m = 3660
.10*15000 + .07*18000 + .075*12000 = 3660
1500 + 1260 + 900 = 3660
3660 = 3660
second equation is true so values for s, b, and m are correct.
-----
s should = (b+m)/2
substitute to get:
15000 = (18000 + 12000)/2
15000 = 30000/2
15000 = 15000
-----
looks good.
-----
i suspect the following statement is where you went wrong.
start of statement:
2x + 2x + x = 45000 (x = stocks with 2x = bonds and money market funds which are twice the amount of stocks)
end of statement:
-----

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Yep.
The total invested in bonds AND money market is twice the amound in stocks, so if x represents the amount invested in stocks, then 2x represents the amount invested in both bonds AND money market funds, therefore:

is the correct first equation. Solving, , so the total invested in stocks was $15,000 and the total invested in bonds AND money market was $30,000.

That means that the total return with respect to stocks was 10% of 15,000 or $1500. It also means that the total return with respect to bonds AND money market funds was $3,660 - $1,500 or $2,160.

Let's now say that the amount invested in bonds is b and the amount invested in the money market is m.

We know two things:

and



Rearranging the first equation we get and then substituting in the second equation we get:



Now simply solve for b and then subtract the value of b from 30000 to get m.

Hope that helps.

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Write a system of equations to solve the following problem. Be sure to identify what the variables represent. Then find the solution.
Ms. Jones purchased a total of $45,000 in stocks, bonds, and money market funds.
Value Equation: s + b + m = 45000
-------------------------------------
The total she invested in bonds and money market funds was twice the amount she invested in stocks.
Value Equation: b+m = 2s
or 2s - b -m = 0
-------------------------------------
The return rates on the stocks, bonds, and money market funds were 10.0%, 7.0%, and 7.5%, respectively. The total value of the return was $3,660.
Interest Equation: 0.10s + 0.07b + 0.075m = 3600
or 10s + 7b + 75m = 360000
-------------------------------------
How much of each investment (stocks, bonds, and money market funds) did Ms. Jones purchase?
------
You have 3 equations with three variables:
s + b + m = 45000
2s - b - m = 0
10s + 7b + 75m = 360000
-------------------------
Solve using any system you know to get:
s = 15000
b = 30000
m = 0000
=============
Cheers,
Stan H.

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B

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Remember, the ordered pair (100,335) means that x=100 and y=335 (or in this specific case C=335)


Since x=100 and C=335, where x is the number of books and C is the cost, this means that 100 books cost $335


So the answer is B.) It costs $335 to produce 100 books.

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you have 4 options.
you need to outline how much each option will cost per week.
assume a week is 7 days without any other information to go on.
they will travel between 400 and 800 miles (given).
option 1:
$329 + $0.0 per mile
option 2:
$219 + $.12 per mile
option 3:
7*$50 = $350 + $0.0 per mile
option 4:
7*$40 = $280 + $.03 per mile
-----
easiest way to do this is to create a function and graph it.
i'll make the functions a,b,c,d
a(x) = 329 + 0*x
b(x) = 219 + .12*x
c(x) = 350 + 0*x
d(x) = 280 + .03*x
graph of all 4 equations follows:
look below the graph for further comments.
negative numbers on the graph are only there so the x and y axis will show up. it won't otherwise.
you can figure out which line is which by looking at the starting values.
on this graph, the x axis is the number of miles traveled and the y axis is the cost.

-----
assuming you will not travel more than 800 miles, the two plans with 0 cost per mile are too expensive relative to the other plans. they only become competitive over 800 miles traveled.
-----
the cheapest plan in the beginning is the one that start off at $219. it's the best up to about 600 miles when the plan that starts off at $280 becomes cheaper.
-----
you could actually solve for the crossover points.
take the plans that start off at $219 and $280.
those are:
b(x) = 219 + .12x
d(x) = 280 + .03x
just make them equal to each other and solve for x.
219 + .12x = 280 + .03x
subtract .03x from both sides:
219 + .12x - .03x = 280
subtract 219 from both sides:
.12x - .03x = 280 - 219
combine like terms:
.09x = 61
divide both sides by .09:
x = 61 / .09 = 677.777777...... miles.
that's the crossover point when both these plans are equal.
it's hard to see on the graph but it's in the general vicinity of 679 miles on the graph.
-----
if you want to plot data points, you may.
takes miles from 400 to 800 in 50 mile chunks and solve for each equation at each of those points.
for example:
1 point is 400 miles
at 400 miles you'll get these numbers:
a(x) = 329 + 0*400 = 329
b(x) = 219 + .12*400 = 219 + 48 = 267
c(x) = 350 + 0*400 = 350
d(x) = 280 + .03*400 = 280 + 12 = 292
-----
you would need to do this every 50 or 100 miles, whichever you prefer. since it's less work, do it every 100 miles.
-----
looking at the graph is easier.
since they're all straight lines, you can actually plot at 400 miles and at 800 miles and draw a straight line in between.
-----

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let x be the number of calories Javier can eat
so since Javier has consumed 1150 already we need to subtract
:
that amount from each limit
:
1500-1150=350 and 1800-1150=650

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so we have 72d+.16m for rate 1 and 144d+.08m for rate 2. where d is days and m is mileage. set is to rate 2 :
144(7)+.08m < 72(7)+.16m
:
1008+.08m < 504+.16m
:
-.08m < -504.......we need to divide by -.08 remembering that when you divide by a negative that the inequality reverses.
:
miles

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You have found the turning point which will give the maximum area correctly.
I would suspect that your calculator is not set to show such a wide range of x and y. Make sure that it will display x up to 200 and the y value up to 10000. The graph should then display.

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We'll go to the Least score which is "90" to get "A" Grade
----->
, EQN 1
, cross multiply


-----> Final score to have "A" Grade
Check it out in EQN 1:



Thank you,
Jojo

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The perimeter of a rectangle is 20 meters. If the length is increased by four
times the width, the sum is 19 meters. What are the dimensions of the rectangle?
What is its area?
:
Original equation
2L + 2W = 20
Simplify divide by 2
L + W = 10; the sum of the length and width of the original rectangle
or
L = (10-W); we can use for substitution
:
"If the length is increased by four times the width, the sum is 19 meters."
(L+4W) + W = 19
L + 5W = 19
:
Substitute (10-W) for L
10 - W + 5W = 19
4W = 19 - 10
4W = 9
W =
W = 2.25; width of the original rectangle
:
Find L:
L = 10 - 2.25
L = 7.75; length of the original rectangle
:
Length increased by 4 times the width
7.75 + 4(2.25) =
7.75 + 9 = 16.75; length of the new rectangle, Width remains at 2.25
:
What is its area?
New rectangle area;
16.75 * 2.25 = 37.6875
:
:
Check the sum of the dimensions of the new rectangle
16.75 + 2.25 = 19

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1. The length of a rectangle is three times the width. The area is 108in^2. Find the length and the width.
The formula for area of a rectangle is:

Plug in the values you already know:

Start by dividing both sides by 3:

Now find the square root of both sides:

To find the length, multiply the width by 3.

Multiply:

So, your width is 6, and your length is 18.

2. The sum of two numbers is six. The sum of the squares of the two numbers is twenty-six. Find the two numbers.

Your two numbers would be 5 and 1.

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Let = the number of pennies
Let = the number of nickels
Let = the number of dimes
Given:
(1) (in cents)
(2)
(3)
------------
From (2),



Substitute in (1)



Multiply both sides by




There are 54 nickels in the jar
Also,



Now check answer:




OK

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lets call the number of pennies p, nickels n, and dimes d.
.01p+.05n+.1d=8.4---eq 1
d=2p-6--------------eq 2
d=n-----------------eq 3
:
lets plug in values from eq 2 and 3 into eq 1 remembering that n=d
.01p+.05(2p-6)+.1(2p-6)=8.4
.01p+.1p-.3+.2p-.6=8.4
.31p=9.3

d=2(30)-6=54
and since d=n
then the number of nickels is 54

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75A+50B=600
3A+ 6B=60------>where A is the number of A cabs and B is the number of B cabs
:
solve for A in the 2nd equation and plug in that value to 1st equation:
:
3A=60-6B--->A=20-2B
:
75(20-2B)+50B=600
1500-150B+50B=600
-100B=-900
B cabinets
3A+6(9)=60--->3A=6
A cabinets

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The width of a rectangle is 3ft less than the length. The perimerter is 62ft. Find the length and the width of the rectangle.
width = x - 3
length = x
Perimeter = 62
62 = 2x + 2(x - 3)
62 = 2x + 2x - 6
62 = 4x - 6
62 + 6 = 4x
68 = 4x
68/4 = x
17 = x
The length is 17 feet.
The width is 3 less than 17 or 14 feet.

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W=L-12
2W+2L=108
2(L-12)+2L=108
2L-24+2L=108
4L=108+24
4L=132
L=132/4
L=33 IS THE LENGTH.
W=33-12
W=21 IS THE LENGTH.
pROOF:
2*21+2*33=108
42+66=108
108=108

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Probably not the way in which your teacher wants you to solve it but it still works.
The average cost of a brisk will be $2.00. If the variable cost is $1.50 then each brick sold will use up $0.50 of the fixed costs.
If the fixed costs are $180.00 and each brick sold uses up $0.50 of that cost, then the number of bricks will be $180.00 / $0.50 = 360 bricks
So the cost of making the 360 bricks will be (360 x $1.50) + $180.00 = $720.00 which gives an average cost of $2.00 per brick.

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Daily cost is $1,000 + $1.50x
ie amount to run factory for the day is $1,000 (fixed cost) and x is the number of cups made costing $$1.50 in materials
If Total = $2,800
then $2,800 = $1,000 + $1.50x
subtract $1,000 to get the x term alone to give
$2,800 - $1,000 = $1.50x
$1,800 = $1.50x
now divide both sides by $1.50 to get x on its own
$1,800/$1.50 = x
so x = # cups made to produce $2,800 daily cost is 1200 cups
Not sure what the 2nd part is asking
but if they want the mugs to be $2.00 each then
If Total = $2,800
then $2,800 = $1,000 + $2.00x
subtract $1,000 to get the x term alone to give
$2,800 - $1,000 = $2.00x
$1,800 = $2.00x
now divide both sides by $2.00 to get x on its own
$1,800/$2.00 = x
so x = # cups made to produce $2,800 daily cost is 900 cups

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In Florida, when there were 447 boats registered (in thousands) there were 13 manatees killed that year. During a year when there were 719 boats registered (in thousands) there were 47 manatees killed that year. Assuming a linear relationship between boats registered and manatees killed, find a linear equation that describes the number of manatees killed with respect to boats registered. (discuss the meaning of the slope for this problem).
-----------------------------
(447,13) and (719,47)
---------------------
slope: (47-13)/(719-447) = 34/272 = 0.125 = 1/8
==================
intercept: 13 = (1/8)447 + b
b = -42.875
--------------------
EQUATION:
# killed = (1/8)(boats in thousands) - 42.875
========================
Meaning of the slope.
When number of boats increases by 1000, number of manatees killed increases
by 1/8
=============
Cheers,
Stan H.

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.15x+.07(50,000-x)=6000
.15x+3,500-.07x=6,000
.08x=6,000-3,500
.08x=2,500
x=2,500/.08
x=$31.250 needs to be invested @ 15%
50,000-31,250=18,750 invested @ 7%
-------------------------------------
.15x+.07(50,000-x)=7000
.15x+3,500-.07x=7,000
.08x=7,000-3,500
.08x=3,500
x=3,500/.08
x=43,750 amount invested @ 15%
50,000-43,750=6,250 invested @ 7%

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dried apples,raisins and peanuts. One pound of the mixture is $3.18
it contains twice as much peanuts by weight than apples.
:
One pound of dried apples cost $4.48, a pound of raisins cost $2.40 and
a pound of peanuts is $ 3.44.
:
How many ounces of each ingredient are contained in one pound of the trail mix?
:
Here it is, step-by-step
:
Let a = no. of oz of apples
Let r = no. of oz of raisins
Let p = no. of oz of peanuts
:
The equation:
4.48a + 2.40r + 3.44p = 3.18(16)
:
It says p = 2a, substitute 2a for p
4.48a + 2.40r + 3.44(2a) = 3.18(16)
:
4.48a + 2.40r + 6.88a = 50.88
:
4.48a + 6.88a + 2.40r = 50.88
:
11.36a + 2.40r = 50.88
:
At this point we can say: r = (16-3a), a represents apples & peanuts here.
Substitute (16-3a) for r, solve for a;
11.36a + 2.40(16-3a) = 50.88
:
11.36a + 38.40 - 7.2a = 50.88
:
11.36a - 7.2a = 50.88 - 38.40
:
4.16a = 12.48
a =
a = 3 oz dried apples
then
P = 2(3)
P = 6 oz of peanuts
:
Find r:
r = 16 - 3 - 6
r = 7 oz of raisins
;
:
Check these values for a, r, p in the cost equation
4.48(3) + 2.40(7) + 3.44(6) = 3.18(16)
13.44 + 16.8 + 20.64 = 50.88; confirms our solutions

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sam has 30 nickles and dimes worth $2.40. How many nickles does he have?
.
Let n = nickles
and d = dimes
.
We have two unknowns, so we'll need two equations:
.
From: "sam has 30 nickles and dimes" we get equation 1:
n+d = 30
.
From: "worth $2.40" we get equation 2:
.05n + .10d = 2.40
.
Using "elimination method":
n+d = 30
.05n + .10d = 2.40
.
multiply the top equation by -.05:
-.05n - .05d = -1.5
.05n + .10d = 2.40
.
Add the two equations together:
-.05n - .05d = -1.5
.05n + .10d = 2.40
------------------------
.05d = .9
d = 18 (number of dimes)
.
Substitute the above into equation 1 to find n:
n+d = 30
n+18 = 30
n = 30-18
n = 12 (number of nickles)

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so we have total cost of let say C
C=130+5(m) where m is the total number of miles divided by 100. 130 is a fixed cost that doesnt change no matter how many miles are driven.
so make a graph where y axis is the cost and the x axis m(number of miles/100)
if miles driven is say 500 then m would be 500/100..so C=130+5(5)=155
so you would plot (5,155) on the graph(perhaps every mark on the graph could equal 5)
when miles driven is 1000, m would be 1000/100=10..so C=130+5(10)=180
(10,180) as you can see for every 500 miles you drive you add $25 to the cost.

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your very close except on the 2nd equation the new mixture needs to be set to
4.50(100) so x+y=100
...........and 5x+3y=4.50(100)
x then =75
y=25

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5.5X+3.50(21-X)=83.50
5.5X+73.5-3.50X=83.50
2X=83.50-73.50
2X=10
X=10/2
X=5 ADULT TICKETS WERE SOLD.
21-5=16 CHILDREN TICKETS WERE SOLD.
PROOF:
5.50*5+3.50*16=83.50
27.50+56.00=83.50
83.50=83.50

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lets call # of cars and motorcycles, c and m respectively.
c+m=28 eq 1
c=3m eq 2
take the value from eq 2 and place it in eq 1.
3m+m=28---->4m=28--->motorcycles
:
cars

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, Working Eqn
Given---
Continuing,



-->
---> ANSWER, binomial(2 terms)
Thank you,
Jojo

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s(t)=-16(t^2)+142t
.
In this case, they are saying that if s(t) equals 172 feet what is t?
.
Simply set s(t) to 172 and solve for 't':
s(t)=-16(t^2)+142t
172=-16(t^2)+142t
16(t^2)-142t+172=0
8(t^2)-71t+86=0
.
Since it is difficult to factor, we use the quadratic equation.
Doing so will yield:
x = {7.428, 1.447}
Units are in seconds
.
Here's the quadratic equation:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=2289 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 7.42771842847642, 1.44728157152358. Here's your graph:

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Charlie owns a coffee shop. He wants to create special mixes using 2 coffees,one priced at $6.40 per pound and the other priced at $7.28 per pound. How many pounds of the $7.28 coffee should he mix with 9 pounds of the $6.40 coffee to sell the mixture for $6.95 per pound?
.
Let x = pounds of $7.28 per pound coffee added
.
7.28x + 6.40(9) = 6.95(x+9)
7.28x + 57.6 = 6.95x + 62.55
0.33x + 57.6 = 62.55
0.33x = 4.95
x = 15 pounds (of $7.28 coffee)

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A high school sold gift wrap. The gift wrap in solid colors sold for $4.00 per roll and the print gift wrap sold for $6.00 per roll. The total amount of rolls sold was 480, and the total amount of money collected was $2,340.00. How many rolls of each kind of gift wrap was sold?
.
Let x = number of $4 rolls sold
then
480-x = number of $6 rolls sold
.
4x + 6(480-x) = 2340
4x + 2880 - 6x = 2340
2880 - 2x = 2340
2880 = 2x + 2340
540 = 2x
270 = x ($4 rolls sold)
.
480-x = 480-270 = 210 ($6 rolls sold)

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Answer: 2c-3

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I am sorry, but I think something is missing in this problem ...at least the way it is written. The cost of the booth is a fixed cost of $312 dollars that vendors must spend in order to operate at a fair and $1410 is a fixed amount that the vendor would like to earn. We can only really generate one equation here with 2 variables as far as I can see
and that would be if A were the number of small milkshakes
............ and B were the number of large milkshakes
then (2.25-.45)A + (3.00-.75)B = $1410+312 (amount you want to make plus fixed costs) so 1.80A+2.25B=1522.
we need more information to solve this in my opinion.
SO MAYBE YOUR TEACHER IS CRAZY EITHER THAT OR I AM....LOL

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I'm sure your teacher is not crazy.:)
Anyway, I'll try and help;
First of all we know he spends $312 all together. Now lets break that down. To make small milkshakes it cost .45 for each, and for large it cost .75 for each.
So the equation would be;
x = small milkshakes
y = large milkshakes
.45x + .75y = 312
Now the lther equation;
We know he wants to make $1410 altogether, he charges 2.25 for small and 3.00 for large.
So the equation;
2.25x + 3y = 1410
Now we have our two equations, lets solve them.
.45x+.75y = 312
2.25x+3y = 1410
========================
let solve the first for y;
.75y = 1410-.45x

y= 416-.6x
Now plug this into either equation for y to find x;
2.25x +3(416-.6x)=1410
2.25x + 1248 - 1.8x=1410
.45x+1248=1410
.45x=162
x=360
Now we have our x, lets find out y;
.45(360)+.75y=312
162 + .75y = 312
.75y = 150

y=200
Check
.45(360) + .75(200)=312
Hope you understand
:)

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Find the three consecutive odd integers such that the product of the first and third integers is 4 less than the square of the second integer
:
The three odd integers: x, (x+2), (x+4)
:
1st*3rd = 2nd^2 less 4
x*(x+4) = (x+2)^2 - 4
:
x^2 + 4x = x^2 + 4x +4 - 4
:
x^2 + 4x = x^2 + 4x; no kidding!
:
No unique solution, any value for x will make the equation happy

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This same problem came up about a week ago, here is what I submitted then
:
The level of thorium in a sample decreases by a factor of one-half every 4.2 million years. A meteorite is discovered to have only 7.6% of its original thorium remaining. How old is the meteorite?
:
The decay formula: Ao*2^(-t/h) = A
Where:
Ao = initial amt
A = resulting amt
t = time (in millions of yrs)
h = half-life of the substance (in millions of years)
:
In this problem: let Ao = 1; A = .076
1*2^(-t/4.2) = .076
:
ln(2^(-t/4.2)) = ln(.076); find the nat log of both sides
:
.693 = -2.577; use the log equiv of exponents
:
= -2.577;
Multiply both sides by 4.2
-.693t = -2.577 * 4.2
:
-.693t = -10.823
t =
t = 15.6 million years old
;
:
Check solution on a calc enter 2^(-15.6/4.2) = .076..


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