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Perimeter = 2L + 2W = 156.

divide by 2 and we get

L = 2W
By substitution, we get


W = 26 meters
L = 52 meters

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To use Cramers rule you need matricies.
step #1 rewrite all equation so that variables are on one side and in order.
5x - 6y - 7Z = 7
6x - 4y + 10z = -34
2x + 4y -3Z = 29.
--
step #2 - We need to find four determinants:
(1) the determinant of the left side, D.
D = -592
(2) the determinant of x, Dx, when you replace the x column with the answer column
Dx = -1184
(3) the determinant of y, Dy, when you replace the y column with the answer column
Dy = -2368
(4) the determinant of z, Dz, when you replace the z column with the answer column
Dz = 1776
--
step #3 use fractions to find the answer:
(1) Dx / D = x answer ; x = 2
(2) Dy / D = y answer ; y = 4
(3) Dz / D = z answer ; z = -3
I don't know an easier way to explain this. Hope this makes sense.

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This is proportions.
280/ 3 = X / 16
cross multiply to get
3X = 4480
X = 1493.33
To the nearest $, X = $1493.

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We need three consecutive even integers. We don't know what they are. Some examples are:
2,4,6 ; 4,6,8 ; 6,8,10 and so on.
--
Notice that the second number is 2 more than the first.
Notice the third number is 4 more than the first.
--
Since we don't know the first number, call is X.
The second number is 2 more than this, it is X + 2.
Third number is 4 more than this, it is X + 4.
--
So, Three consecutive even integers can be expressed as: X, X+2, X+4.
Two times the smallest is the largest means
(i) 2X = X + 4,
since the smallest was X and the largest was X + 4.
subtract X from both sides to get
X = 4.
Now
the first number was X. It is X = 4,
the second number was X + 2, it is 4 + 2 = 6
the third number is X + 4, it is 4 + 4 = 8.
The three numbers are 4 , 6, 8

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Let's look at restrictions first:
X can't = 1, 2, or -5
If X equals any of these the log wouldn't exist at log(0).
--
step #1 - place all front numbers in exponents (power rule)

--
step #2 - add of logs means multiply information and subtract means divide information. SO,
log
---
step #3 - simplify the numerator. This is not the fun step . . .
log

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This is an RTD problem. Here is the table
vehicle . . . . . . . rate . . . . . . . . time . . . . . . . .distance
cycle . . . . . . . . . R . . . . . . .. . . 270/R . . . . . .. 270
cycle . . . . . . . . .R + 15 . .. . . . 270/R - 3 . . . .. 270
I let R = rate. The first time is D / R = 270 / R. All times are in hours, so the second time is
(270/R - 3) hours.
Look at the second line.
(R + 15)(270/R - 3) = 270
By FOIL, we get
270 - 3R + 4050/R - 45 = 270
multiply both sides by R. we get
270R - 3R^2 + 4050 - 45R = 270R.
Simplify to get
3R^2 + 45R - 4050 = 0
dividing by 3, we get
R^2 + 15R - 1350 = 0
factoring, we get
(R + 45)(R - 30) = 0
R = -45 or R = 30.
we can use -45, so
R = 30. The motorcycle was traveling at 30 mph.

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Volume is
V = LxWxH
V = (X)(2x+3)(2x+5)
V = 4x^3 + 16x^2 + 15X.

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This is a mixture problem. Here is the table:
SALINE . . . . . % . . . . . .gal . . . . . .%gal
A . .. . . . . . .. 20 . . . .. . G . . . . . . . 20G
B . . . . . . . . .. 65 . . . . .120-G . . . .7800 - 65G
mix . . . . . . . 50 . . . . . .120 . . . . . .6000.
--
I put in all information from the problem and then I had to use variables. I let G = gallons of A and 120-G be gallons of B. It is always total gallons - X.
Now,
20G + 7800 - 65G = 6000
-45G = -1800
G = 40.
I need 40 gallons of the 20% saline solution and 80 gallons of 65% saline solution.

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First rectangle.
a Rectangle is twice as long as it is wide translates to
(i) L = 2W
If its length and width are both dimentions are increased by 4 m is translated to
L+4, W + 4.
its area is increased by 88m^2 means that new area - original area = 88m^2.
--
original area from (i)
A = LW = (2W)(W) = 2W^2
--
new area after + 4 to both L and W.
A = LW = (2W+4)(W+4) = 2W^2 + 12W + 16.
--
new area - old area = 88 is
2W^2 + 12W + 16 - 2W^2 = 88
12W + 16 = 88
12W = 72
W = 6
L = 12.
I am hoping from this you can make the sketch.

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Two equations with two unknowns have the possible type of solutions:
(1) no solution - parallel lines
(2) 1 solution - lines cross at one point.
(3) infinite solutions - the lines are the same but in disguise.

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bird . . . .35 = r
bird . . . 65 = r + c
bird . . . 5 = r -c
substitute r with 35.
65 = 35 + c
c = 30.
check
5 = 35 - 30 [check]
the speed of the wind is 30 mph

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Price of shirt = P.
total cost including tax is
P (1 + tax rate). So,
45(1 + .05)
45*(1.05)
$47.25

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Let's start with the first equation and then repeat process for second:
(i)
STEP #1: factor out the 3 from the first two terms to get
(ii)
--
STEP #2: take 1/2 middle term number and square it. Multiply it by 3 and put that answer in the blanks. We get
(iii)
--
STEP #3: write the left side as a quantity squared and equalling the right.
(iv)
This is vertex form.
--
--
(v) y= -2x^2+28x-7
STEP #1: factor out -2 from the first two terms to get
(ii)
--
STEP #2: take 1/2 middle term number and square it. Multiply it by -2 and put that answer in the blanks. Sometimes it's easier to say [middle number/2]. Don't forget your signs. We get
(iii)
--
STEP #3: write the left side as a quantity squared and equalling the right.
(iv)
This is vertex form.

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We have the coordinates
A (-3,3) and B(1,-8)
to find midpoint, we need the following formula:
,
putting in our numbers, we get
(-3 + 1)/2 , (3 + -8)/2 - -> (-1, -5/2)
To find length, we need distance formula, which is

again putting your numbers in, we get



d ~ 11.70 ~ 11.7

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We have the following table:
Gas used: 5, 8, 12, 15, 18, 20, 22, 25
Miles driven, 80, 100, 180, 225, 280, 290, 320, 375.
To create a linear model, we want to apply some rule to x and then get y.
Notice
5*15 = 75 + 5 = 80
8*15 = 120 - 20 = 100
12*15 = 180
15*15 = 225
18*15 = 270 + 10 = 280
20*15 = 300 - 10 = 290
22*15 = 330 - 10 = 320
25*15 = 375.
We can't create a perfect linear model, but this means we have a "regression" model, or an equation that gets us close:
M = 15G.
Hope this helps.

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graph
FIrst i would reduce where possible. We get
y = (-1/9)x + (1/2)
Your slope = -1/9 and b value, y intercept is 1/2.
STEP #1 - place a dot at 1/2 on the Y-axis.
STEP #2 - from this dot, you will move down 1 unit and right 9 units. PLace a second dot. connect the dots. and you are done.

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let's start with the first equation and then move to the second:

quadratics are expressed as Y = Ax^2 + Bx + C.
--
vertex:
We can find the vertex using -b/2a. -5/2*10 = -5/20 = -1/4. This is the x part of the vertex and now we put that into f(x) to get
10(1/4)^2 + 5(1/4)-1 = 10/16 + 5/4 - 1 = 14/16 = 7/8.
vertex = (1/4,7/8)
--
axis of symmetry:
This is just the x value of the vertex
x = 1/4.
--
min:
since the graph opens up, there will be a min at the vertex.
min @ (1/4,7/8)
--
domain:domain: For a parabola opening up or down, domain will be all reals.
range: Since the parabola opens up, the minimum range is 7/8. So we have Y >= 7/8.
-- EQUATION #2
{{y = -24x^2 + 18x + 11}}
quadratics are expressed as Y = Ax^2 + Bx + C.
--
vertex:
We can find the vertex using -b/2a. -18/2*-24 = 18/48 = 3/8. This is the x part of the vertex and now we put that into f(x) to get
-24(3/8)^2 + 18(3/8) + 11 = -27/8 + 27/4 + 11 = 115/4.
vertex = (3/8, 115/4)
--
axis of symmetry:
This is just the x value of the vertex
x = 3/8.
--
min: No min in this case.
max: SInce the graph opens down, there will be a max at the vertex.
min @ (3/8, 115/4)
--
domain: For a parabola opening up or down, domain will be all reals.
range: Since the parabola opens down, the highest range is 115/4. So we have Y <= 115/4.

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Let's take them one at a time:
---
2.
(i) y = 3 - x
(ii)y = x + 1
Creating a table of values, we get:
X . . . -2 . . -1 . . . 0 . . . 1 . . . 2
Y(i). . .5 . . . 4 . . . .3 . . . 2 . . . 1
Y(ii) . .-1 . . 0 . . . 1. . . .2 . . . 3
Notice the Y's are equal at 2, when X = 1. You should be able to graph this now.
---
3.
(i) y = 1/2 x -3/2
(ii) y = 2x-6/4
Creating a table of values, we get:
X . . . . .. . . -4 . . . . . . .-2 . . . . . . . . . 0 . . . . . . . . 2 . . . .. . . . 4
Y(i). . . . . -7/2 . . . .. . -5/2 . . . . . . . -3/2 . . .. . .-1/2 . . . . . 1/2
Y(ii) . . . .-19/2 . . . . . . -11/2 . . . . . -3/2 . . .. . . .5/2 . . . . . . 13/2
Notice the Y's are equal at -3/2, when X = 0. You should be able to graph this now.
---
4.
(i)y = x+3/2
(ii)y = 2x-3/4
Creating a table of values, we get:
X . . . . .. . . 0 . . . . . . .3/4 . . . . . . . . . 6/4 . . . . . . . . 9/4 . . . .. . . . 12/4
Y(i). . . . . 3/2 . . . .. . 9/4 . . . . . . . . . 3 . . . . . . . 15/4 . . . . . . . .9/2
Y(ii) . . . .-3/4 . . . . . . 3/4 . . . . . . 9/4 . . . . . . . . .15/4 . . . . . . . 21/4
Notice the Y's are equal at 15/4, when X = 9/4. You should be able to graph this now.

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OUr original equations are:
(i)
(ii)
--
STEP #1 - multiply (i) equations by 5
(iii)
(ii)
--
STEP #2 - add down to eliminate by.
(v) 28ax = 56ab.
We must assume that a, b, x, and y are not all zero.
x = 2b.
-- STEP #3 - find y.
Y = 3a.

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Let's start with the first equation and then repeat process for second:
(i)
STEP #1: re write the equation as
(ii)
--
STEP #2: take 1/2 middle term number and square it. put that answer in the blanks. We get
(iii)
--
STEP #3: write the left side as a quantity squared and equalling the right.
(iv)
This is vertex form.
--
--
(v) y = x^2 - 9x + 3
STEP #1: re write the equation as
(ii)
--
STEP #2: take 1/2 middle term number and square it. put that answer in the blanks. Sometimes it's easier to say [middle number/2]. Don't forget your signs. We get
(iii)
--
STEP #3: write the left side as a quantity squared and equalling the right.
(iv)
This is vertex form.

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This is what I call job / time. Here is the formula:
J1/T1 * (together time) + J2/T2 * (together time) = total number of jobs.
J1 is firs persons number of jobs
T1 is first persons time to do the jobs.
Two machines fill cereal box. So, J1 = J2 = 1; B total cereal boxes.
T1 = T2; same rate.
At this pint we have the following equation:
(1/t)(4) + (1/t)(4) + (1/t)(26) = B.
We have to add the alone time of the second machine after the first breaks down. We don't know how many total boxes were to be filled.
Continuing
8/t + 26/t = B
34 = Bt
t = 34/B
This gives you the time to fill B boxes.
Hope this helps.

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The equation for company 1 is
C = 4S, where C = total cost and S = cost per shirt.
The equation for company 2 is
C = 3S + 75.
The cost is the same when both equations are equal to each other. So,
4S = 3S + 75
S = 75; the answer
---
The check:
If S = 75, the cost for company 1 is
C = 4*75 = $300
If S = 75, the cost for company 2 is
C = 3*75 + 75 = 225 + 75 = $300.

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Step #1 - solve 3x + 4y -8 = 0 for y.
Y = -(3/4)X +2.
STEP #2 - find the slope based on the equation from STEP #1. M = (-3/4).
STEP #3 - parallel lines have the same slope. The slope you have from STEP #2 is the slope you want. m = (-3/4).
STEP #4 - using the slope from STEP #3, pick a point (2,3) and put those into
Y = mX + b. Solve for b.
3 = (-3/4)(2) + b.
b = 9/2.
STEP #5 - write the equation.
Y = (-3/4)X + 9/2.

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Here is the original equation:
(i) f(x) = 3^(2 - 3x)
We want
first, [f(1 + x)]
wherever you see x replace it with 1 + X. We get
f(1 + X) = 3^(2 - 3*(1 + X))
(ii) f(1 + X) = 3^(-3X - 1)
Second, [f(1 - x)]
wherever you see x replace it with 1 - X. We get
f(1 - X) = 3^(2 - 3*(1 - X))
(iii) f(1 - X) = 3^(3X - 1)
Now, we multiply (ii) and (iii) to get
(iv) [f(1 + X)][f(1 - X)] = 3^(-3X - 1) * 3^(3X - 1)
Since the bases are the same, we just add the exponents. We get
(v) [f(1 + X)][f(1 - X)] = 3^(-2) = 1/9.

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Here is the question
(i) 2/3%of ?=33.3
rewrite the percent as / 100.
2/300 * X = 33.3
---
Now we solve.
multiply both sides by 300.
2X = 9990
divide by 2.
X = 4995.

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We are given this polynomial:
p(x) = x^4 + 7x^3 + 9x^2 - 17x - 20.
Using PNI, we get
P . . . N . . . I
1 . . . 3 . . . 0
1 . . . 1 . . . 2.
So, I know there MUST be 1 positive root.
Now I look at P/Q; all the factors of 20 over all the factors of 1.
+-20 +-10 +-5 +-4 +-2 +-1.
I use synthetic division until I find a hit: (X + 4) and (X + 1) are factors of P(x).
So, (X + 4)(X + 1)[X^2 + 2X - 5] = P(x).
The zero's become
X = -4, -1, -1 + sqrt(6), -1 - sqrt(6).

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We need the Pythagorean theorem:
(i) a^2 + b^2 = C^2.
There are really three choices:
If a^2 + b^2 < C^2 then obtuse
If a^2 + b^2 = C^2 then right
If a^2 + b^2 > C^2 then acute
---
The numbers given must be expressed from least to greatest: sqrt(8), sqrt(66), and sqrt(73). Put them into (i) and find out which of the three we get.
(sqrt(8))^2 + (sqrt(66))^2 ? (sqrt(73))^2
8 + 66 ? 73
74 > 73.
So we say acute.

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The pipe is broken into two pieces. Let the first piece be X and the second piece is 9X.
X + 9X = 50.
10X = 50
X = 5.
One piece is 5 feet and the other piece is 45 feet.

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Here is the original equation:
5[42-(10-90)]
---
Using order of operations, 10-90 first.
5[42-(-80)]
---
a double negative is positive. So,
5[42+80]
---
Now, we simplify the (_)
5[122]
---
Finally multiply to get 610.

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Let X = first number and Y = second number.
The sum of two numbers is -64 can be translated as
(i) X + Y = -64
Twice the first number is 1 more than the second can be translated as
(ii) 2X = Y + 1.
We can use substitution to solve. Solve the (ii) for Y. We get
(iii) Y = 2X - 1
Substitute (iii) into (i) to get
(iv) X + 2X - 1 = -64
Solve for X.
3X - 1 = -64
3X = -63
X = -21
Find Y.
Y = -43.

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You are given the following
7>-4x+6 or 10≤-5x+4.
We first solve the left side:
7 > -4x + 6
1 > -4X
-1/4 < X; notice the switch of inequality when dividing by (-).
Now, we solve the right side:
10 ≤ -5x + 4
6 ≤ -5x
X ≤ -(6/5); notice the switch of inequality when dividing by (-).
So, the answer I have is:
(-∞,-6/5] u (-1/4,∞).