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7 a raised to 3 b raised to 2,
:
Assuming this is what you mean here:
=
:
when a=1/2 and b=1/3:
:
= = = = 12.25
:
:
If you mean:
=
:
when a=1/2 and b=1/3:
:
= = = = 1.75

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Three years ago, Milcah is 3 times older than Joshua. 4 years from now, Milcah is 8 years older than Joshua. Find Joshua's and Milcah's age.
:
M = M's age now
J = J's age now
:
Write what it says:
"Three years ago, Milcah is 3 times older than Joshua."
(M-3) = 3(J-3)
M - 3 = 3J - 9; multiplied what's inside the brackets
M = 3J - 9 + 3
M = (3J-6)
:
"4 years from now, Milcah is 8 years older than Joshua."
M+4 = (J+4) + 8
M + 4 = J + 12
M = J + 12 - 4
M = J + 8
:
Find Joshua's and Milcah's age.
Substitute (3J-6) for M in the above equation:
3J - 6 = J + 8
3J - J = 8 + 6
2J = 14
J = 14/2
J = 7 yrs; J's age now
:
Find M:
M = J + 8
M = 7 + 8
M = 15 yrs; M's age now
:
:
Check solution using the statement:
"Three years ago, Milcah is 3 times older than Joshua."
(15-3) = 3(7-3)
12 = 3(4); confirms our solution

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x + y + 2z = 11
x + y + 3z = 14
x +2y + 3z = 5
:
Look at it; you can see that subtracting the 1st eq from the 2nd, you eliminate x & y making it easy to solve for y:
x + y + 3z = 14
x + y + 2z = 11
------------------ Subtracting eliminates x & y
0 + 0 + 1z = 3
z = 3
:
Substitute 3 for z in the 2nd and 3rd equations:
2nd equation:
x + y + 3(3) = 14
x + y + 9 = 14
x + y = 14 - 9
x + y = 5
:
3rd equation
x + 2y + 3(3) = 5
x + 2y + 9 = 5
x + 2y = 5 - 9
x + 2y = -4
:
Using these two unknown equations:
x + 2y = -4
x + y = 5
------------subtracting eliminates x
0 + 1y = -9
y = -9
:
Find x using the equation x + y = 5
x - 9 = 5
x = 5 + 9
x = 14
:
We have x = 14; y = -9; z = 3
:
Check solution in the 1st equation: x + y + 2z = 11
14 - 9 + 2(3) =
14 - 9 + 6 = 11
:
Did this give you a clue, at least?

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Nikki and tatianna rollerblade in opposite direction. Tatianna averages 2 mph faster than nikki. If they began at the same place and ended up 20 mi apart after 2 hrs, how fast did each of them travel?
:
Let s = N's speed
Then
(s+2) = T's speed
:
the total distance they will travel is 20 mi, write a distance equation.
Distance = time * speed
:
N's dist + T's dist = 20 mi
2s + 2(s+2) = 20
:
2s + 2s + 4 = 20
:
4s = 20 - 4
:
s = 16/4
:
s = 4 mph is N's speed
The
4 + 2 = 6 mph is T's speed
:
:
Check solutions by finding the distance of each:
2*4 = 8 mi
2*6 = 12mi
------------
total 20 mi

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The diameter of a small pizza is 16 centimeters. This is 2 centimeters more than two fifths of the diameter of a large pizza. Find the diameter of the larger.
:
Let x = diameter of the large pizza:
= .4
:
.4x + 2 = 16
.4x = 16 - 2
.4x = 14
x = 14/.4
x = 35 cm is the diameter of the large pizza

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you invested $8000 in two funds paying 2% and 5% annual interest respectively. At the end of the year, the interest from the 5% investment exceeded the the interest from the 2% investment by $85. How much money was invested at each rate?
:
Let x = amt invested at 5%
Then
(8000-x) = amt invested at 2%
:
A simple equation
Interest at 5% - interest at 2% = $85
.05x + .02(8000-x) = 85
:
.05x - 160 + .02x = 85
:
.05x + .02x = 85 + 160
:
.07x = 245
:
x = 245/.07
:
x = $3500 invested at 5%
Then
8000 - 3500 = $4500 invested at 2%
:
Check solution by find the interest on both amts
.05(3500) = 175
.02(4500) = 95
----------------
Difference = 85 as given

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If (x,y) is a point on the line l of slope -2 and y intercept 2, y = -2x + 2. (So, for example, if (23,y) is a point on l, y= -2(23)+2 = -44.)
:
Assume you understand what they are saying here:
:
:
k is the line that is perpendicular to l and contains the point (18,13). If (x,y) is a point on k, express y in terms of x.
:
Perpendicular lines have slope relationship that can be expressed:
m1 * m2 = -1
Assume m1 = -2 (line l slope), and find m2 (k line slope which is perpendicular)
-2*m2 = -1
:
m2 =
:m2 = + is the slope of k line
:
We know the slope of k so find the y intercept (b), by using x,y of 18,13
Using the slope intercept form
y = mx + b
13 = 18 + b
13 = 9 + b
13 - 9 = b
b = +4
Our k equation is:
y = x + 4
:
Confirm this, substitute 18 for x and see that y = 13
y = (18) = 4
y = 9 + 4
y = 13
:
If you graphed these two lines, it would look like this:

purple line is l, y intercept at +2
green line is k, y intercept at +4
:
Did this help you understand this stuff somewhat?

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A plane took 1 hour longer to travel 560 miles on the first portion of a flight than it took to fly 480 miles on the second portion. If the speed was the same for each portion, what was the flying time for the second part of the trip?
:
Let t = time required for the 2nd part of the trip
Then
(t+1) = time required for the 1st part
:
It said the speed was the same, write a simple speed equation:
Speed = Dist/time
:
1st part speed = 2nd part speed
=
Cross multiply and you have
560t = 480(t+1)
560t = 480t + 480
560t - 480t = 480
80t = 480
t = 480/80
t = 6 hrs for the 2nd part of the trip
:
Check solution by finding the speed of each portion of the trip;
1st part required 7 hrs
= 80
= 80
:
Kind of slow for a plane, but it checks out.

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Two friends live 60 miles apart. They ride their bikes to meet each other. Sally heads east at 21 mph. Therese heads west riding at a rate of 15 mph. How far does each cyclist ride in t hours. Write and solve an equation to find when they meet.
:
Let t = time in hrs when they will meet
:
We know that when they meet they will have traveled a total of 60 mi.
From this fact we can make a simple distance equation: Distance = Speed * time
:
S's dist + T's dist = 60 miles
21t + 15t = 60
36t = 60
t = 60/36
t = = 1 hrs or 1 hr 40 min
:
:
Find how far each will travel and check solution for t:
S's dist + T's distance = 60 miles
21* + 15* =
35 miles + 25 miles = 60 miles

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Graph the linear equation 3x-2y=6
Put the equation in the general "y= "form:
3x - 2y = 6
-2y = 6 - 3x
:
We always want a positive y here, mult equation by -1
2y = 3x - 6
:
We want y to have a coefficient of 1, divide eq by 2
y = x - 3
:
Plot at least two points.
Here's three points: Substitute for: x = -2, x = -2 and x = +4, you should have:
X | Y
--------
-2 | -6
+2 | 0
+4 | +3
:
The graph will look like this:

:
Can you do this now?

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: Cashews sell for $4.00 per pound, Peanuts for $1.50 per pound. If you mix 60 lbs of peanuts in a cashew/peanut mix, how many pounds of cashews do you need to add to make a mix that costs $2.50 per
:
Let x = lb of cashews required
Then
The final amt = (x+60)
:
A nutty equation:
4x + 1.5(60) = 2.5(x+60)
:
4x + 90 = 2.5x + 150
:
4x - 2.5x = 150 - 90
:
1.5x = 60
:
x = 60/1.5
:
x = 40 lb of cashews required
:
:
Check solution in original equation:
4(40) + 1.5(60) = 2.5(40+60)
160 + 90 = 2.5(100) proves our solution
:
:
Not that hard, right?

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Impossible, N will always be older than A

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The length of a rectangle is 9 centimeters more than it's width.
This says: L = (W+9)
:
The perimeter is 82 centimeters.
you know: 2L + 2W = 82
;
Substitute (W+9) for L, and find W:
2(W+9) + 2W = 82
:
Simplify, divide equation by 2 and you have:
W + 9 + W = 41
2W = 41 - 9
2W = 32
W = 32/2
W = 16 is the width
:
It said the length was 9 cm more: L = 25 cm
:
:
Check the perimeter now
2(25) + 2(16) = 82

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x^2 + y^2 = 16
y^2 = 16 - x^2
y =
:
xy = 8
substitute for y, find x:
x* = 8
Square both sides
x^2(16-x^2) = 64
:
-x^4 + 16x^2 - 64 = 0; a quadratic equation
+x^4 - 16x^2 + 64 = 0; multiplied by -1
Factor:
(x^2 - 8)(x^2 - 8) = 0
x^2 = +8
x =
x =
:
Find y:
*y = 8
y =
y =


(x + y)^2 = ?
FOIL
(+)*(+) = ?
4(2) + + + =
8 + 8 + 8 + 8 = 32

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A fence is to be built around a track so that there is a uniform gap between the outside edge of the track and the fence. The Length of the straight portion of the track is 100 meters and the diameter of the curve is 81 meters. What is the Maximum width of the gap between the track and the fence if no more than 777 meters of fencing is used?
:
Let x = width of the gap
:
If you draw this out, it is apparent that 200 m of fencing will run parallel to
the straight portions of the track. Therefore, 777 - 200 = 577 ft is available
for the two half circles. The radius of the two half circles will be (x+81).
:
Use the circumference formula:
2 * pi * (x+81) = 577
6.2832x + 508.938 = 577
6.2832x = 577 - 508.938
6.2832x = 68.062
x = 68.062/6.2832
x = 10.83 ft is the width between the fence and the track
:
Did this make sense? Any questions?

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My average electricity bill for the months of November, December, January, and February was $200. The highest month was January which was twice February. December was $10 less than January and November was $24 less than February. How much was my bill in each of the months November, December, January, and February?
:
= 200
:
Write equations for each statement:
:
"January which was twice February."
J = 2F
or
F = .5J
:
"December was $10 less than January"
D = J - 10
:
"November was $24 less than February."
N = F - 24
We want to get everything in terms of J, substitute .5J for F:
N = .5J - 24
:
Substitute and you have;
= 200
:
Mult by 4 to get rid of the denominators, remove the brackets:
.5J - 24 + J - 10 + J + .5J = 800
:
.5J + J + J + .5J - 24 - 10 = 800; group like terms and do the math
3J - 34 = 800
3J = 800 + 34
J = 834/3
J = $278
Then
F = 278/2 = $139
and
N = 139 - 24 = $115
and
D = 278 - 10 = $268
:
:
Check on a calc:
= 200

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Solve:
:
(x + 5)^2 = 3
Take the square root of both sides:
x + 5 = +-
:
Subtract 5 from both sides
x = -5 + sqrt(3)
and
x = -5 - sqrt(3)

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he demand and supply equations for a certain item are given by
D = –5p + 40
S = –p^2 + 30p – 8
Find the equilibrium price.
:
The equilibrium price (p) occurs when D = S, therefore:
:
-5p + 40 = -p^2 + 30p - 8
:
Arrange as a quadratic equation on the left:
+p^2 - 5p - 30p + 40 + 8 = 0
:
p^2 - 35p + 48 = 0
:
This equation requires the quadratic formula to solve: a = 1; b =-35; c= 48

:

:

:

:

Two solutions


p = 33.57
and


p = $1.43
:
Only the lower price make sense, the higher value will give negative values in the original expressions.
:
You can check this by substituting 1.43 for p
-5(1.43) + 40 = -(1.43^2) + 30p - 8
-7.15 + 40 = -2.0449 + 30(1.43) - 8
32.85 = -2.0449 + 42.9 - 8
32.85 = 32.85 equality reigns
:
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A small company produces both standard and deluxe playhouses. The standard playhouses take 12 hours of labor to produce, and the deluxe playhouses take 20 hours. The labor available is limited to 800 hours per week, and the total production capacity is 50 items per week. Existing orders require the company to produce at least 10 standard playhouses and 15 deluxe playhouses per week. Write a system of inequalities representing this situation, where x is the number of standard playhouses and y is the number of deluxe playhouses. Then graph the system of inequalities.
:
Let x = standard p.h; y = deluxe p.h
:
The labor equation:
12x + 20y =< 800
:
put equation in the general (y=) form to plot the graph:
20y =< 800 - 12x
y =< 800/20 - (12/20)x
y =< 40 - .6x
:
:
The production capacity equation:
x + y =< 50
Put this in the "y=" form also
y <= 50 -x
:
Existing order constraints
x => 10
and
y => 15
:
Plot these using the equation givens.
y = 40 - .6x; (purple line)
y = 50 - x; (green line)
y = 15; Note that y = 15 is a horizontal line going thru y = 15; black line
x = 10 is a vertical line going thru x = 10, I am unable to put this in
:
I assume you know how to make up an x/y table and plot a graph, if you can't,
let me known and we will go thru the graphing routine
:
Here is the graph:

:
When you draw your graph (or print this one) draw a vertical line thru x = 10
This will complete the area of feasibility which is:
:
1.At or below, the purple line or the green line whichever is lower.
2.At or above the black horizontal line
3.At or to the right of the vertical line which you will add to the graph
:
Does this all make sense to you?

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A rectangular plot, bounded on one side by a river, is to be enclosed on the other 3 sides by a fence, and then divided into two equal sized pens by another fence. If you have 900 feet of fence available, what is the largest are that can be enclosed?
:
Perimeter would be:
L + 3W = 900
or
L = (900-3W)
:
Area would be:
A = W * L
Substitute for L
A = W(900-3W)
A = -3W^2 + 900W
:
A quadratic, find the axis of symmetry using: x = -b/(2a); a = -3; b= 900
W = -900/(2*-3)
W = -900/-6
W = 150 ft is the width for max area
:
Find Max area, substitute 150 for W
A = -3(150^2) + 900(150)
A = -3(22500) + 135000
A = -67500 + 135000
A = +67500 sq ft is max area
:
If you graph it: x = width, y = area

:
:
Check it:
L = 900 - 3(150)
L = 900 - 450
L = 450 ft
So it would be 450 by 150: 450*150 = 67500 sq ft
:
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Premium coffee sells for $6.00 per pound and regular coffee sells for $4.00 per pound. how many pounds of each type of coffee should be blended to obtain 100 pounds of a blend that sells for $4.64 per pound.
:
Let x = lb of premium coffee
It says the total will be 100, therefore:
(100-x) lb of regular coffee:
:
A simple equation:
6x + 4(100-x) = 4.64(100)
:
6x + 400 - 4x = 464
:
6x - 4x = 464 - 400
:
2x = 64
:
x = 64/2
:
x = 32 lb of premium coffee
:
You can figure out how much regular coffee, I'm sure

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40 = (2)(1+g)/ 13.8-g
:
I am assuming you mean:
= 40
:
= 40; multiplied what's inside the brackets
:
2 + 2g - 13.8g = 13.8(40); multiplied eq by 13.8, gets rid of the denominator
:
2g - 13.8g = 552 - 2; Subtracted 2 from both sides
:
-11.8g = 550; do the math
:
g = 550/-11.8; divide both sides by -11.8
:
g = -46.61 is the solution
:
:
Check in original equation:
= 40
= 40; minus a minus is a plus
= 40;
= 40;
-6.61 + 46.61 = 40; confirms our solution
:
:
If this is NOT the way you meant the problem to be, let us know, we can edit this solution

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Two cars pass each other traveling in opposite directions on a highway, one is going 1.5 times as fast as the other. At the end of 3.5 hours, they are 192.5 miles apart. Find each cars rate.
:
Let s = speed of the slower car
Then
(1.5s) = speed of the faster car
:
Write a distance equation: Distance = time * speed.
:
They both will be traveling for 3.5 hr
;
Fast car dist + slow car dist = 192.5
3.5(1.5s) + 3.5s = 192.5
5.25s + 3.5s = 192.5
8.75s = 192.5
s = 192.5/8.75
s = 22 mph is the slow car
Then
1.5(22)) = 33 mph is the fast car
:
:
Check solution:
3.5(33) + 3.5(22) =
115.5 + 77 = 192.5: confirms out solution

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Use elimination to solve:
x/3 + 4y = 30
2x - y = 5
:
Multiply the 1st equation by 6 to get rid of the denominator, and
giving us a 2x to use for easy elimination, results:
2x + 24y = 180
2x - y = 5
----------------subtracting eliminates x, find y;
0x + 25y = 175
25y = 175
y = 175/25
y = 7
:
Find x using 2x - y = 5
2x - 7 = 5
2x = 5 + 7
2x = 12
x = 12/2
x = 6
:
Check solutions using 1st equation
6/3 + 4(7) =
2 + 28 = 30; confirms our solution
:
:
Use elimination to solve:
6x - y/4 = -9
3x + 2y = -30.
:
Multiply 1st equation by 8, eliminates the fraction and
gives a 2y for easy elimination:
48x - 2y = -72
3x + 2y = -30
----------------adding eliminates y, find x
51x + 0y = -102
51x = -102
x = -102/51
x = -2
:
Find y, using 3x + 2y = -30
3(-2) + 2y = -30
-6 + 2y = -30 + 6
2y = -24
y = -24/2
y = -12
:
Check solution in 1st equation:
6(-2) - (-12/4) =
-12 + 3 = -9
:
:
Solve the system of equations by substitution:
y = x^2 + 2x - 2
y = 3x + 4
:
Substitute (3x+4) for y in the 1st equation:
x^2 + 2x - 2 = 3x + 4
x^2 + 4x - 3x - 2 - 4 = 0
x^2 + x - 6 = 0; a quadratic equation:
Factor:
(x-3)(x+2) = 0
x = +3 and x = -2
Find y using the 2nd equation, when x = +3
y = 3(3) + 4
y = 13
when x = -2
y = 3(-2) + 4
y = -2
:
You can check both solutions by substituting for x in the 1st equation:
:
:
Mr.Adams buys a bag of apples and a bag of pears for a total of $14.00.There are eight pounds of fruit in all.The apples cost $1.50 per pound,and the pears cost $2.30 per pound.How many pounds of pears did he buy?
:
Let a = lbs of apple, and p = lbs of pears
The total lb equation:
a + p = 8
or
p = (8 - a); use for substitution:
:
The total cost equation:
1.50a + 2.30p = 14.00
Substitute (8-a) for p in the above equation:
1.5a + 2.3(8-a) = 14
1.5a + 18.4 - 2.3a = 14
1.5a - 2.3a = 14 - 18.4
-.8a = -4.4
a = -4.4/-.8
a = + 5.5 lb of apples
Find p using the lb equation:
p = 8 - 5.5
p = 2.5 lb of pears
:
Check solution in the cost equation:
1.5(5.5) + 2.3(2.5) =
8.25 + 5.75 = 14; confirms our solutions
:
How about this, are you understanding what's going on here?
:

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A ball is thrown upward from the roof of a building 100 m tall with an initial velocity of 20 m/s. When will the ball reach a height of 80 m?
:
This can be made into a quadratic equation with the three elements
Let t = time in seconds
1. Gravity (negative because it's accelerated downward) -4.9t^2
2. Initial velocity (positive because it's thrown up-ward: + 20t
3. Initial height above ground: +100
:
-4.9t^2 + 20t + 100 = height above ground
:
They want to know when will it be at a height of 80 ft:
-4.9t^2 + 20t + 100 = 80
:
-4.9t^2 + 20t + 100 - 80 = 0; subtracted 80 from both sides
:
-4.9t^2 + 20t + 20 = 0
:
This is the quadratic formula: except we will be solving for t instead of x:

Substitute: a=-4.9; b=20; c=20

Do the math:

; minus a minus is a plus
:

:

Solution 1


t = -.83, this is NOT our solution
:
Solution 2:


:
t = + 4.9 sec, this IS our solution

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A 60 by 80-foot rectangular walk in a park is surrounds a flower bed. If the walk is of uniform width and its area is equal to the area of the flower bed, how wide is the walk.?
:
Total area: 60 * 80 = 4800
Since the walk and the flower bed have equal area,
Flower bed area = 2400 sq ft
:
Let sidewalk width = x
:
Dimension of the flower bed: (60-2x) by (80-2x)
Therefore:
(60-2x) * (80-2x) = 2400
:
4800 - 280x + 4x^2 = 2400; FOILed
:
4x^2 - 280x + 4800 - 2400 = 0
:
4x^2 - 280x + 2400 = 0; a quadratic equation
:
Simplify, divide by 4:
x^2 - 70x + 600 = 0
:
Factors to:
(x-60)(x-10) = 0
:
x = 10 ft is the width of the sidewalk (reasonable solution)
:
:
Check solution by finding out if the area is 2400
:
(60-20)*(80-20) = 2400 sq/ft
:
:
Did this make sense to you? Any questions?

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Esteemed Student,
On this site you have to submit a specific problem to be explained by a tutor.
Seldom will anyone answer a general request like that. You are encouraged to ask
questions about the problem you are having difficulty with. If you have done some
work on it. Include that too. Hope to hear from you. A

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Mary, Sue and Bill work at a motel. If each worked alone, it would take Mary 10 hours, Sue 8 hours, and Bill 12 hours to clean the whole motel. One day Mary come to work early and she had cleaned for two hours when Sue and Bill arrived and three finished the job. How long did they take to finish.?
:
Let t = time it took the 3 of them to finish the job
Let the completed job = 1
:
M came early so she did of the competed job:
:
M alone + three together = completed job
+ + + = 1
:
Multiply equation by a common denominator, 120 would work:
120* + 120* + 120* + 120* = 120(1)
:
Cancel out the denominators and you have:
12(2) + 12t + 15t + 10t = 120
:
24 + 37t = 120
:
37t = 120 - 24
:
t = 96/37
:
t = 2.6 hrs for the 3 to finish the job
:
Check solution on a calc:
M worked 2 + 2.6 = 4.6 hrs:
Mary + Sue + Bill = completed job
= 1.00 hrs, confirms our solution
:
Did this make some sense to you? Any questions?

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-9(x-3)^2 = -7
:
1(x-3)^2 = ; divided both sides by -9
:
(x-3)^2 =
:
x - 3 = +/-; extract the square root of 1/9
:
x = 3 +
and
x = 3 -
:
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Rectangle of a perimeter is 240cm. If the length of rectangle is decreased by 10% & breadth is increased by 20% the perimeter remain same(i.e. 240cm) find the length & breadth of rectangle
:
"Rectangle of a perimeter is 240cm."
:
Let x = the original length; y = original width
:
2x + 2y = 240
Simplify divide equation by 2:
x + y = 120
or
y = (120-x); for substitution
:
"If the lenght of rectangle is decreased by 10% & breadth is increased by 20% the perimeter remain same(i.e. 240cm)"
New length = .9x
New width = 1.2y
:
2(.9x) + 2(1.2y) = 240
Simplify, divide by 2
.9x + 1.2y = 120
:
"find the length & breadth of rectangle"
:
Substitute(120-x) for y in the above equation, find x:
.9x + 1.2(120-x) = 120
.9x + 144 - 1.2x = 120
.9x - 1.2x = 120 - 144
-.3x = -24
x = -24/-.3
x = + 80 cm is the original length
:
Width: 120 - 80 = 40 cm is the original width
:
Find the new dimension:
.9 * 80 = 72 cm, new length
1.2 * 40 = 48 cm new width
:
Check the new perimeter
2(72) + 2(48) =
144 + 96 = 240

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Doomstown is 200 miles due west of Sagebrush, and Joshua is due west of Doomstown. At 9 A.M. Mr Archer leaves Sagebrush for Joshua. At 1 P.M. Mr. Sassoon leaves Doomtown for Joshua. If Mr. Sassoon travels at an average speed 20 mph faster than Mr Archer and they reach Joshua at 4 pm, how fast is each traveling?
:
Josh-----------------d--------------Doom----------------200------------Sage
:
A's travel time: 9am to 4pm = 7 hr
S's travel time: 1pm to 4pm = 3 hr
:
A's speed = s
S's speed = (s+20)
:
Let d = distance from Doom to Josh; (Distance = time * speed)
:
d = 3(s+20); (S's speed times 3 hr)
d = 3s + 60; (use for substitution)
:
(d + 200) = 7s; (A's speed times 7 hrs)
:
In the above equation, Replace d with (3s+60) and find s:
(3s + 60 + 200 = 7s
3s + 260 = 7s
260 = 7s - 3s
260 = 4s
s = 260/4
s = 65 mph is A's speed
Then
65 + 20 = 85 mph is S's speed
:
:
Check solution by finding d
d = 3s + 60
d = 3(65) + 60
d = 195 + 60
d = 255 mi
:
255/85 = 3 hrs S's time
(200+255)/65 = 7 hrs, A's time