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*
:
*
:
Factor out (a-y) in the denominator
*
:
Cancel (a-y) & (b+3)
*
:
Cancel 3 into 6
*
:
Use -1 to make (b-3)
*
:
Cancel (b-3)
*
:
-1 * =

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: how can i solve this word problem?:
one number is 4 less than another number. the sum of the two numbers if four times their difference. what are the two numbers?
:
Two numbers: x & y
;
Just write an equation for what it says:
:
"one number is 4 less than another number."
x = y - 4
:
" the sum of the two numbers is four times their difference."
x + y = 4(x - y)
x + y = 4x - 4y
y + 4y = 4x - x
5y = 3x
:
what are the two numbers?
From the 1st statement, replace x with (y-4) in the above equation:
5y = 3(y - 4)
5y = 3y - 12
5y - 3y = -12
2y = -12
y = -6
;
Find x
x = -6 - 4
x = -10
;
:
Is this true, does 5y = 3x
5(-6) = 3(-10)
:
Check in the statement:
the sum of the two numbers is four times their difference."
-10 - 6 = 4(-10 -(-6))
-16 = 4(-10 + 6)
-16 = 4(-4)

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John says to Brenda, "If you give me $7, we will have an equal amount of money." Brenda responds, "That may be true, but if you give me $7, I will be $1 short of having twice as much money as you." How much money did John and Brenda have?
:
Let j = amt that John has
Let b = amt that Brenda has
;
Step-by-step
J says to B,"If you give me $7, we will have an equal amount of money."
j + 7 = b - 7
j = b - 7 - 7
j = (b - 14)
;
B says, "if you give me $7, I will be $1 short of having twice as much money as you."
b + 7 = 2(j-7) - 1
b + 7 = 2j - 14 - 1
b + 7 = 2j - 15
b = 2j - 15 - 7
b = 2j - 22
:
How much money did John and Brenda have?
From the 1st statement replace j with(b-14) in the above equation
b = 2(b-14) - 22
b = 2b - 28 - 22
b = 2b - 50
+50 = 2b - b
b = $50 is Brenda's amt
Find j
j = b - 14
j = 50 - 14
j = $36 is John's amt
;
:
Check this in both statements
J says to B,"If you give me $7, we will have an equal amount of money."
36 + 7 = 50 - 7
43 = 43
and
B says, "if you give me $7, I will be $1 short of having twice as much money"
50 + 7 = 2(36-7) - 1
57 = 2(29) - 1
57 = 58 - 1

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What is the vertex of the graph of y = |2x - 3| - 5? What is the range of the function?
You should be able to recognize that the value inside the absolute has to be positive or 0,
therefore the the range (y) has to be - 5 or greater, right?
:
Plot a couple points here
x = -3
y = |2(-3) - 3| - 5
y = |-6 - 3| - 5
y = |-9| - 5
Remove the absolutes
y = 9 - 5
y = +4
Plot x/y point -3, +4
:
Plot the value so that the value inside the absolute = 0
x = 1.5
y = |2(1.5) - 3| - 5
y = |3 - 3| - 5
y = 0 - 5
y = -5
Plot point x/y : 1.5, -5, the minimum range
;
x = +3
y = |2(3) - 3| - 5
y = |6-3| - 5
y = 3 -5
y = -2
Plot points x/y +3, -2
:
Looks like this:

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Consider the function y - 5 = -4x
:
a) Write the equation in standard form.
The standard form is ax + by = c
y - 5 = -4x
add 4x to both sides to get 4x on the left
4x + y - 5 = 0
add 5 to both sides to get the numeral (c) on the right, results
4x + y = 5; this is the standard form
:
b) Write the equation in slope-intercept form.
The slope intercept form is y = mx + b
y - 5 = -4x
Add 5 to both sides
y = -4x + 5; this is the slope/intercept form
:
c) The slope is: in the above equation note that the slope (m) is -4
:
d) The y-intercept is 5 (b).
:
e) Graph the function using the slope-intercept method of graphing.
:
Using the slope/intercept equation y = -4x + 5
Substitute a value for x and find y
x = -1
y = -4(-1) + 5
y = +4 + 5
y = 9
Plot this x/y point: -1,9
:
x = +1
y = -4(1) + 5
y = -4 + 5
y = 1
Plot this x/y point: 1,1
;
this is linear graph (straight line) we only need to plot the two points
the graph should look like this:

Note that this agrees with the two points, -1,9 and 1,1
Also see the y intercept occurs when x = 0, then y = 5
;
:
Did this shed some light on this stuff for you??
Question? ankor@att.net
:
Response to comment:
That does present a problem but can be overcome by plotting a point that falls
within this graph

you can see we are limited but lets plot the point for:
x = +2
y = -4(2) + 5
y = -8 + 5
y = -3
Plot this x/y point: +2, -3
:
So we have 2,-3 and 1, 1 as the points we can plot here
Also I should mention the y intercept is an easy one to plot, (when x=0), the
point where the line crosses y axis, in this case 5, can be used
:
You can choose what values of x you wish to plot, if one doesn't work try
another, with a little practice you will see what is best.
:
Did this help answer your question on this? Let me know please.

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Assume the problem is:

Get rid of the denominator, multiply both sides by 9, results:
-4(2x - 4) = 9(48)
Multiply what's inside the brackets by -4, results:
-8x + 16 = 432
:
-8x = 432 - 16
:
-8x = 416
x has to be positive, multiply both sides by -1
8x = -416
Divide both sides by 8
x =
x = -52
;
:
Check solution in original equation, replace x with -52



Cancel 9 into 108
-4(-12) = 48
+48 = 48; confirms our solution
:
:
:
Did you understand this?

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solve xy=6
If you solve for x, divide both sides by y and you have
x =
:
If you solve for y, divide both sides by x and you have
y =

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A rancher in Calgary decides to build a silo inside his corral.
The rectangular corral measures 22 m by 11 m and the silo's base is circular with a diameter of x meters.
Find a polynomial for the remaining area of the corral (in square meters) after the silo is built.
:
The area of the corral: 22 * 11 = 242 sq/m
:
The area of the base of the silo: ( radius is half the diameter)
A =
:
Remaining area = Corral area - silo base area
A(x) = 242 -
:
A(x) = 242 -
or
A(x) =

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I am 13 years old, and my coach is 31 years old, which is my age with the digits
reversed.
What is the fewest number of years in which the digits of our ages will be reversed again?
:
The difference ages = 18
:
Let x = First digit of your age
Let y = 2nd digit of your age
:
You are 13, therefore a simple equation
y = x + 2
:
Substitute values for x
when x = 1, y = 3; 13, 31 the original ages
when x = 2, y = 4; 24, 42 is the next time the ages are reversed
:
Note that difference between reversed numbers is always multiples of 9

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three machines A , B and C take 3 hours to do a job, when working together.
Suppose A and B are identical and that C working alone can complete the entire job in 12 hrs.
How long would it take A working alone to do the job?
:
Let t = time for A or B working alone (Identical machines)
:
Let the completed job = 1
:
+ + = 1
:
+ = 1
Multiply equation by 12t, results
12(6) + 3t = 12t
:
72 = 12t - 3t
:
72 = 9t
t =
t = 8 hrs, A working alone
;
:
Check solution
+ + =
+ = 1

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Travis paddles his kayak in the harbor at Morro Bay, California, where the incoming tide has caused a current in the water.
From the point where he enters the water, he paddles 2 miles against the current, then turns around and paddles 2 miles back to where he started.
His average speed when paddling with the current is 3 miles per hour faster than his speed against the current
If the complete trip (out and back) takes him 1.4 hours, find his average speed when he paddles against the current.
:
s = av speed against the current
then it says, av speed with the current is 3 mph faster than against, therefore
(s+3) = av speed with the current
:
Write a time equation: time =
Time with + time against = 1.4 hrs
+ = 1.4
Multiply equation by s(s+3), results:
2s + 2(s+3) = 1.4s(s+3)
:
2s + 2s + 6 = 1.4s^2 + 4.2s
:
4s + 6 = 1.4s^2 + 4.2s
;
Arrange as a quadratic equation
1.4s^2 + 4.2s - 4s - 6 = 0
:
1.4s^2 + .2s - 6 = 0
:
Use the quadratic formula to find the solution for s

In this equation x=s; a=1.4; b=.2; c=-6

:

:

:

the positive solution

s =
s = 2 mph against the current
:
:
Check solution by finding the times. (speed with the current: 2+3 = 5 mph}
+ = 1.4

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I need to find the individual totals for 3 months that when added together, equal 88,635.
However, each month has a different total which reflects a .0043 percent of increase from that month, to the next month.
Ex. (Month 1 increases by .0043 to get Month 2) (Month 2 increases by .0043 to get Month 3) and all three months equal 88,635.
:
Change .0043% to a decimal .000043
:
let x = month one total
then
1.000043x = month two
and
1.000043(1.000043x) = month three
:
month 1 + month 2 + month 3 = 88635
x + 1.00043x + 1.000043(1.000043x) = 88635
:
x + 1.00043x + 1.000086x = 88635
:
3.000129x = 88635
:
x =
x = $29,543.73 is month one total
then
1.000043(29543.72) = $29,545.00 is month two total
and
1.000043(29545) = 29546.27 is month three total
;
:
Check by finding the total of the three months.
29543.73 + 29545.00 + 29546.27 = 88635

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we purchased 60 units and gave away three and sold the rest at twice their purchase price. The total profit was $1188 how much did we sell them for?
;
Let x = purchase price
then
2x = selling price
:
Revenue - cost = profit
2x(60-3) - 60x = 1188
:
2x(57) - 60x = 1188
:
114x - 60x = 1188
54x = 1188
x = 1188/54
x = $22 the purchase price
then
$44 price they sold them for
:
:
Check this:
44(57) - 22(60) =
2508 - 1320 = 1188

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A cylindrical can is to have volume 300 cubic centimetres. Find its height and its
radius if the total surface area (and hence the total amount of material used) is minimum.
[The volume is given by V = r2h, and the surface area is S = 2rh + 2r2, where r is the radius and h is the height.
:
Find h in terms of the Volume
= 300
h =
:
Surface area
S = +
Replace h with
S = +
we can cancel pi*r
S = +
S = +
:
Plot the equation :

With the help of a TI83, minimum on this graph: x=3.628, (Min SA ~ 248 sq/cm)
:
hence r = 3.628 cm is the radius
:
Find the height
h = = 7.255 cm is the height
:
:
Check this by finding the volume with this r and h
V =
V = 300.00

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2. If Sally can paint a house in 4 hours, and John can paint the same house in 6 hour,
how long will it take for both of them to paint the house together?
:
Here's an easy method to do these "shared work" problems
:
Let t = time required when they work together
:
Let the completed job = 1; (a painted house in this case)
:
Each person will do a fraction of the work, the two fractions add up to 1
+ = 1
multiply equation by 12 to get rid of the denominators, results:
3t + 2t = 12
:
5t = 12
t =
t = 2.4 hrs working together
Convert .4 hrs to min: .4 * 60 = 24 min
t = 2 hr 24 min
:
:
You can prove this to yourself:
+ =
.6 + .4 = 1
:
Wasn't that pretty easy to understand??

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Replace the blanks in the equation with constants to complete the square and form a true equation.
x^2 + 2/3x + ____ = (x + ___)^2
:
To complete the square, divide the coefficient of x by 2, and square it
1/2 * 2/3 = 2/6, (2/6)^2 = 4/36, so we have:
:
x^2 + 2/3x + 4/36 = (x + 2/6)^2
:
You can reduce the fractions
x^2 + 2/3x + 1/9 = (x + 1/3)^2
:
:
you can prove this to yourself
FOIL (x + 1/3)*(x + 1/3)

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The amount of time it takes a swimmer to swim a race varies inversely as the average speed of the swimmer.
A swimmer finishes a race in 400 seconds with an average speed of 3 feet per second.
Find the average speed of the swimmer if it takes 300 seconds to finish the race.
:
you can use an inverted ratio equation: let s = speed of the faster swimmer
:
=
which is
=
cross mult
3s = 4*3
3s = 12
s =
s = 4 ft/sec is the faster swimmers speed
:
:
Prove that to yourself.
the dist for both should be the same:
400 * 3 = 1200 ft
300 * 4 = 1200 ft

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Joanne left the driveway 40 seconds after Gloria driving down a winding four mile country road.
If Joanne's speed averaged 25 miles an hour what would Gloria's average speed be for Joanne to catch up to her after the third mile?
:
Let s = Gloria's speed to catch Joanne
;
Let this happen at 3.1 miles
:
Change 40 sec to hrs
= .011 hrs
:
Write a time equation (Time = )
;
J.s travel time = G's travel time + 40 sec (.011 hrs)
= + .011
:
multiply equation by 25s
3.1s = 3.1*25 + .011(25s)
:
3.1s = 77.5 + .275s
3.1s - .275s = 77.5
2.825s = 77.5
s =
s = 27.43 mph
:
:
Check by finding the times
3.1/25 = .124 hrs
3.1/27.43 = .113 hrs
----------------------
time diff = .011 hrs which is: .011 * 3600 ~ 40 sec

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Weight varies directly with gravity.
A Mars lender weighed 767 pounds on Earth but only 291 pounds on Mars.
Its Accompanying Mars rover weighed 155 pounds on Mars.
How much did it weigh on Earth? Round your answer to the nearest pound.
:
Use a ratio equation, let x = Rover weight on earth
:
=
Cross multiply
291x = 155 * 767
:
291x = 118885
x =
x = 408.539 ~ 409 lb on earth
:
:
Check with calc
291/767 = .379
155/409 = .379

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Roast Beef has 25g or protein and 11g of calcium per serving.
A serving of mashed potatoes has 2g of protein and 25g of calcium.
How many servings of each are needed to supply exactly 29g of protein and 61g of calcium?
:
let b = servings of beef
let p = servings of potatoes
:
Write two equations, one for protein and one for calcium
:
Protein
25b + 2p = 29
:
Calcium
11b + 25p = 61
:
Multiply the 1st equation by 25, multiply the 2nd by 2
625b + 50p = 725
22b + 50p = 122
-------------------subtraction eliminates p
603b = 603
b = 1 serving of beef
:
Find p
25(1) + 2p = 29
2p = 29 - 25
2p = 4
p = 2 servings of potatoes
;
;
check solution in the calcium equation
11(1) + 25(2) = 61

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A picture’s length is 6in less than twice its width.
A uniform frame of 2in surrounds the picture.
The area of the picture and frame together is 352in^2.
Find the dimensions of the picture.
:
Let x = the picture's width
then
(2x-6) = the picture's length (6" less than twice the width)
;
The 2" frame adds 4" to each dimension, therefore
x + 4 = overall width
and
(2x-6) + 4 = (2x-2) = overall length
;
The overall area equation
(x+4)*(2x-2) = 352
FOIL
2x^2 - 2x + 8x- 8 = 352
:
2x^2 + 6x - 8 - 352 = 0
:
2x^2 + 6x - 360 = 0
Factors to
(2x+30)(x-12) = 0
:
Positive solution
x = 12" is the width of the picture
and
2(12)-6 = 18" is the length of the picture
;
:
Check solutions by finding the total area:
(18+4) * (12+4) = 352

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how to write a quadratic equation with a solution of 5.
:
How about
(x-5)^2
FOIL
(x-5)*(x-5) = x^2 - 10x + 25

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Pairs of markings a set distance apart are made on highways so that police can detect drivers exceeding the speed limit.
Over a fixed distance, the speed R varies inversely with time T.
In one particular pair of markings, R is 45 mph when T is 6 seconds.
Find the speed of a car that travels the given distance in 5 seconds.
:
Speed varies inversely with time so we can write an inverted ratio equation, find r
=
Cross multiply
5r = 6 * 45
:
5r = 270
r =
r = 54 mph if he passes the markers in 5 sec

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A metallurgist needs to make 12.4 lb. of an alloy containing 50% gold.
He is going to melt and combine one metal that is 60% gold with another metal that is 40% gold.
How much of each should he use?
:
Since 50 is half way between 60 and 40 it would be half and half, 6.2, 6.2
But here is the method for any mixture
:
Let x = amt of 60% gold required
then since the resulting amt is 12.4
(12.4-x) = amt of 40% gold
:
Write an "amt of gold" equation, using the decimal equiv of %
.6x + .4(12.4-x) = .5(12.4)
:
.6x + 4.96 - .4x = 6.2
.6x - .4x = 6.2 - 4.96
.2x = 1.24
x =
x = 6.2 lb of 60% gold

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rewrite 7/10 and 2/3 so that they have a common denominator; then determine if they are<,=, or >
;
The common denominator would be 3*10 = 30
:
>

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If the ratio of the length to the width of a painting is "the golden ratio" ,
find the length, in feet, of a painting that has a width of 14 feet.
:
Let x = the length
:
=
Cross multiply
2x =
divide both sides by 2
x =
x =
Using a calc
x ~ 22.652 feet is the length

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If Bennett had a certain number more nickels and fewer dimes, he would have .60 less.
What is the number?
:
Let x = the number
nickel value - dime value = .60 less
+.05x - .10x = -.60
-.05x = -.60
x =
x = +12 is the number
;

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Find the circumference of a circle 3.75 ft.
Use 3.14 pi and round your answer to one decimal place.
:
If 3.75 ft is the diameter, then you just multiply it by pi:
3.14 * 3.75 = 11.774 rounded to 11.8

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What two-digit positive integer is one more than a multiple of 2, 3,4, 5 and 6 ?
:
Primes representing all these numbers: 2, 2, 3, 5,
:
2 * 2 * 3 * 5 = 60 + 1 = 61

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2 cars leave intersection, one travel east; other north.
When the car traveling east had gone 15 miles the distance between the cars was
5 miles more than the distance traveled by the car heading north, how far had
the north bound car traveled.
:
this is a right triangle problems a^2 + b^2 = c^2
:
Let a = 15; the distance by the Eastbound car
Let b = distance traveled by the Northbound car
:
The distance between the cars is the hypotenuse
"distance between the cars was 5 miles more than the distance traveled by
the car heading north,", therefore
c = b+5
:
The equation:
15^2 + b^2 = (b+5)^2
:
FOIL the right side
225 + b^2 = b^2 + 10b + 25
:
Subtract b^2 from both sides
225 = 10b + 25
:
Subtract 25 from both sides
225 - 25 = 10b
200 = 10b
b =
b = 20 miles the distance of he northbound car
;
:
See if this is true
a=15
b=20
c=25
:
15^2 + 20^2 = 25^2
225 + 400 = 625!
:
How about this? Did it make sense to you?

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: if two angles of a triangle are 30 degrees and 50 degrees what is the measure of the third angle?
:
The total of all three angles in any triangle is 180:
Therefore; 180 - 30 - 50 = the third angle