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Assuming you started with
3a +b/7 + b = 3a/7
Multiply both sides by 7 to get rid of fractions.
21a +b +7b = 3a
Subtract 3a from both sides
21a +b +7b -3a = 0
Collect terms
18a +8b =0
Divid by 2
9a +4b = 0

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With problems like this, you have to keep track of how much "pure" stuff you need.
It says you need 50 lb of 50% nickel.
50*.5 = 25 lbs of pure nickel
.
The total of 50 lb is divided between the amount of 40% nickel and 80% nickel
x = the amount of 40% nickel
(50-x) = the amount of 80% nickel
.
.4x + .8(50-x)=25
.4x + 40 - .8x = 25
-.4x = -15
x = 37.5 lb of 40% alloy
.
50-x = 12.5 lb of 80% alloy
.
Check your answer.
.4*37.5 = 15
.8*12.5 = 10
15+10 = 25
.
Done.

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With coin problems you have to keep track of the number of coins and of their value.
For example,
d = number of dimes
10d = value of the dimes in cents
n = number of nickels
5n = value of the nickels in cents
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Given
Aviva has $4.65, which is 465 cents.
Aviva has only nickels and dimes, so we can restate the value.
5n + 10d = 465
Aviva has 57 coins in total.
n + d = 57
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So we can treat these as a system of equations.
.
5n + 10d = 465
n + d = 57
Multiply the second equation by 5
.
5n + 10d = 465
5n + 5d = 285
.
Subtract the second equation from the first equation
5d = 180
Divide both sides by 5
d = 36
.
Substitute
n + d = 57
n +36 = 57
n = 57-36
n = 21
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Check to see if this is right.
.
5n = 5*21 = 105
10d = 10*36 = 365
105+365 = 465
which is the right answer
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Of course, you could have solved using substitution.
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5n + 10d = 465
n + d = 57
n = 57-d
5(57-d) +10d = 465
285 -5d + 10d = 465
As you can see, you get to the same answer, just in a different way.
.
Done.

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A nonlinear equation is one in which x has a power.
For example

.
Note that a nonlinear equation is one that cannot be a straight line.
The following is the graph of


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The following is the graph of


.
The following is the graph of


.
Notice that the number of curves (or changes in direction) is one less than the power of x.

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Get the equation in y=mx+b slope-intercept form.

Subtract x from both sides

Divide both sides by 3

or

.
When x=0, y=0: (0,0)
When x=3, y=-1: (3,-1)
Plot the points and draw the line.
.

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3x^2 +5x = 0
The first two factors share an x.
x(3x +5)=0
.
One root is x=0
The other is found by solving 3x = -5
x = -5/3
.
Done

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The question is missing...

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Given:


.
Substitute what you are given.

Multiply both sides by 100.

Equals are equal, so you can swap sides.

Divide both sides by 180.


.
So, the $60 million would result in the removal of 33.33333% of the pollutants.
.
Done.

.

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3x^2 + 5x = 0
x(3x+5) = 0
.
So, one root is x=0
.
3x+5 = 0
3x = -5
x = -5/3
.
That's another root.
.

.

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You could try: 9, 12, 15.
Check to see if they fit the Pythagorean Theorem's requirements.
C^2 = A^2 + B^2
A = 9
B = 12
C = 15
.
A^2 = 81
B^2 = 144
A^2+B^2 = 81+144
81+144 = 225
sqrt(225) = 15
.
That's correct.
.
Done.

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A line perpendicular to another line with have negative inverse slope.
The define original line is
7x - 10y = 18
Subtract 7x from both sides
-10y = -7x + 18
Divide both sides by -10
y = 7/10*x -18/10
This equation is in slope-intercept format
y = mx +b
m = 7/10
So, the perpendicular line will have slope
-10/7
.
y = -10/7*x + b
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You need the line to go through the point (14,-2).
That means when x=14, y=-2
Substitute
that with what we already know
-2 = -10/7*(14) + b
We can cancel the 14 with the 7
-2 = -10(2) + b
-2 = -20 + b
Add 20 to both sides
18 = b
.
y = -10/7*x + 18
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Always check your answer, so substitute x=14 and see what value of y you get.
y = -10/7*14 + 18
y = -20 + 18
y = -2
That's right.
.
A graph is always helpful. In this case, graph both equations to see if they're perpendicular by visual inspection (of course, you KNOW they're perpendicular because of the slopes)...

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Always list what you know first.
p = perimeter
l = length
w = width
p = 2l + 2w
p = 2(l+w)
p = 90
.
The given relationship has to be turned from English into algebra.
.
length is 3 cm longer than twice the width
l = 2w + 3
.
Substitute and solve for l.
.
p = 2l + 2w
p = 2(2w+3) + 2w
p = 4w + 6 + 2w
90 = 6w + 6
84 = 6w
6w = 84
w = 84/6
w = 14 cm
.
Substitute this value to find l.
.
l = 2*w + 3
l = 2*14 + 3
l = 31 cm
.
Check the perimeter using
l = 31
w = 14
2(31)+2(14) = ??
62 + 28 = 90
Right.
.
Done

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The standard distance formula is:
d = rt
where
d = distance
r = rate (or speed)
t = time
.
r = 600
d = 1200
.
So,
d = rt
Divide both sides by r
d/r = t
t = d/r
t = 1200/600
t = 2
.
Done

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The slope is the change in y divided by the change in x, also called "rise over run".
.
The two points are (2,4) and (-1,-2).
.
Change in y is:
-2 -4 = -6
.
Change in x is:
-1 -2 = -3
.
Slope = -6/-3 = 2
.
Done.

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Start by defining what their ages are now.
J = Jeffrey
M = Michael
.
Three years ago means -3.
Jeffrey was twice as old as Michael is now.
J-3 = 2M
.
In nine year means +9.
J+9 = 4M
.
Set up the linear equations.
J -2M = 3
J - 4M = -9
..
Subtract the second from the first.
2M = 12
M = 6
.
Substitute.
J-3 = 12
J = 15
.
Always check your work!
In this case, use the second equation.
J+9 = 4M
J+9 = 15+9 =24
4M = 4*6 = 24
Right.
.
Answer: Jeffrey is 15, and Michael is 6.
.
Done.

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The basic distance equation is
d = rt,
where
d = distance
r = rate
t = time
.
Define
s = speed of the plane
w = speed of the wind
.
Flying with a tailwind
r = s+w
.
Flying with a headwind
r = s-w
.
Flying with the tailwind
d = 2400
t = 4.75
2400 = 4.75*r
r = 2400/4.75
r = 505.263158
r = 505.26
.
Flying with the headwind
d = 2400
t =6
2400 = 6*r
r = 400
.
s + w = 505.26
s - w = 400
Add
2s = 905.26
s = 452.63
.
Find w.
s+w = 505.26
w = 505.26 - 452.63
w = 52.63
.
Check to see if the distances work.
.
6*(452.63-52.63) = ??
6*(400) = 2400
Right
.
4.75*(452.63+52.63) = ??
4.75*(505.26) = 2399.985
which is close enough to 2400.
.
Answer: The speed of the plane in still air is 452.6 mph.
.
Done.

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The basic distance equation is:
d = rt
.
We are told that flying with the tailwind
900 = r*3
r = 300
BUT
r = speed of the plane + speed of the tailwind
.
We are told flying with the headwind
300 = r*2
r = 150
BUT this time,
r = speed of the plane - speed of the wind
.
Assuming the wind is constant...which is important.
.
We can define:
s=speed of the plane
w=speed of the wind
.
Substitute what we know:
s+w = 300
s-w = 150
.
Add the linear equations.
2s = 450
.
Solve for s
s = 225
.
Substitute to solve for w
w = 75
.
Check by substituting these value to see if you get the correct distance.
.
3(225+75) = ??
3(300) = 900
Right
.
2(225-75) = ??
2(150) = 300
Right
.
Re-read the question to see what answer is required.
.
Answer: The plane's speed in still air is 225 miles/hr.
.
Done.

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P = perimeter
L = length
W = width
P = 2(L+W)
.
P = 20
.
L = W+2
.
(W+2)*(W) = 20
.
W^2 + 2W = 20
.
Done.

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y = mx + b
.
When x=0, y = 4 (given)
(0,4)
.
When y=0, x = 1 (given)
(1,0)
.
Recall that the change in y divided by the change in x = slope.
.
Change in y = 4-0 = 4
Change in x = 0-1 = -1
Slope = 4/-1 = -4
.
The equation is thus:
y = -4x +4
.
The graph is:

.
Notice that the y-intercept is (0,4), the x-intercept is (1,0), and the line goes through both points.

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There is no solution unless you either know the interior angle, or you know it is a right triangle.
The Pythagorean Theorem can be used.
.
Assuming the 13 is the hypotenuse,
.






.
Assuming the hypotenuse is unknown.
.





.
Done.

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L * W = 108 sq ft
Length is 6 ft greater than twice the width.
L = 2W + 6
Substituting
(2W+6)*W = 108
2W^2 + 6W = 108
Divide both sides by 2
W^2 + 3W = 54
W^2 + 3W - 54 = 0
Factor
(W +9)(W -6) = 0
Roots are W=-9 and W=6.
A negative width is impossible, so W=6
L = 2W + 6
L = 2(6) + 6
L = 18
.
Always check your answer.
L*W = ??
18 * 6 = 108
Correct.
.
Done

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s = speed of car going west
s+10 = speed of the car going east
t = 1 hr 36 min
t = 1.6 hr
d = 184 miles between the two cars
.
But they do not travel 184 miles each. They meet somewhere in the middle, but not exactly in the middle because they are going different speeds. However, their combined distance traveled = 184.
.
1.6s + 1.6*(s+10) = 184
3.2s +16 = 184
3.2s = 168
s = 52.5 is the speed of the car going west
s+10 = 62.5 is the speed of the car going east
.
Always check your work.
.
In this case we have to check how far they travel to check these values.
.
1.6 hr * 62.5 mph = 100 miles
1.6 hr * 52.5 mph = 84 miles
.
So, the speeds of the two cars are 62.5 and 52.5 mph.
.
Done.

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factor
2x^2 +x -3 = 0
.
(2x + ?)(x - ?) = 0
.
Well the factors of 3 are 1 and 3.
So, does that work or do we have to use the quadratic equation?
.
(2x+3)(x-1) = 0
.
So the solutions are x=1 or -3/2.
.
The graph is

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s = lengths of the sides of the square
s^2 = area
(s+7)^2 = 121 given
(s+7)(s+7) = 121
s^2 +7s +7s + 49 = 121
.
s^2 +7s +7s + 49 -121 = 0
s^2 +14s -72 = 0
.
factor
(s + ?)(s - ?) = 0
.
2*36 = 72, difference = 34, nope.
4*18 = 72, difference = 14, which is it!
.
(s +18)(s -4) = 0
.
s = -18 or 4
a negative length does not make sense, so s = 4
.
add 7 and check area to make sure it is 121
.
11*11=121 is true.
.
The sides of the original square were 4 inches.
.
Done.

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In week 1, Ed purchased stock:
x = number of shares of stock he bought
p = purchase price per share
xp = 2000
.
One week later...the stock tumbled...
price is now (p-10)
he could have bought 10 more shares: (x+10)
for the same purchase price
(x+10)(p-10)=2000
.
since x*p = 2000
x = 2000/p
p = 2000/x
.
(x+10)(p-10) = 2000
xp -10x +10p -100 = 2000
.
substitute for x
(2000/p)*p - 10*(2000/p) +10p -100 = 2000
2000 -20000/p +10p -100 = 2000
-20000/p +10p = 100
multiply by p
-20000 + 10p^2 = 100p
10p^2 -100p - 20000 = 0
divide by 10
p^2 -10p - 2000 = 0
factor
20*100 = 2000 < 40*50 = 2000 < .
(p-50)(p+40) = 0
.
p = 50 or -40, but a negative stock price is nonsense.
.
p = 50
x = 2000/p
x = 40
.
So, in week 1, Ed purchased 40 shares at $50 per share.
The next week the stock fell by $10
Now the price is $40.
For $2000 he could buy 50 shares now.
Which is 10 more shares than he bought the first week.
.
Done.

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y-intercept occurs when x=0, which is the point (0,50)
-x^2+5x+50 can be factored
(-x+10)(x+5)=-x^2 -5x +10x +50
So, x=10 and x=-5 will be the roots.
y=0, so the points will be (-5,0) and (10,0)
A graph is always a good idea.

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5/2x + 1/4x = 11/4 + x
.
These fractions are painful. But we can treat them all as quarters.
.
Recall
5/2 = 10/4
Also,
x = 1/1x = 2/2x = 3/3x = 4/4x...
.
10/4x + 1/4x = 11/4 + 4/4x
.
11/4x = 11/4 + 4/4x
.
Subtract 4/4x from both sides
7/4x = 11/4
.
Multiply both sides by 4
7x = 11
.
Divide both sides by 7
x = 11/7
.

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x + 2y = 4
2y = -x +4
y = -x/2 + 4/2
y = -x/2 + 2
.
4 + 4y = 8
4y = 8 -4
4y = 4
y = 1
.

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This is a peculiar question because the number of choices does not matter. You either get the answer right or wrong. 60% of 5 = 3/5, so you need to get 3 of the 5 questions right.
.
You can get them Right or Wrong in several ways, but do you need to know that, too?
.
RRRWW
.
Shift left and wrap the one that went of the left end onto the right end
RRWWR
.
Repeat
RWWRR
.
Repeat
WRRRW
.
If you repeat again, you're right back to where you were, so that is not a "new" sequence.
.
Or you could start with RWRWR and go through those
RWRWR
WRWRR
RWRRW
WRRWR
RRWRW
RWRWR is a repeat sequence
.
Or you start with: WRWRR, or WWRRR, etc.
But that is a lot of work, and error-prone, too.
.
If you have done anything with permutations and combinations, you may recognize that the question is asking how many ways you can pick 3 specific items from a set of 5 items: C(n,r), where n=5, r=3



Ignoring the "*1" factors



That means there are 10 ways you can get 3 questions right out of 5 total questions.
.
Now if the question were to have asked "at least 60%" then the answer would be different because you would have to consider the case of getting all 5 right as well as 4/5 right.
.
Done

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.



.



Adding we get...

.
Answer:

.
Done

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Recognize that, in this case, "of" means multiply:
1/3 * 60 = 20
.
Similarly,
25% = 1/4
So,
1/4 * 60 = 15
.
The "difference" means to subtract, which is
20 - 15 = 5
.
Done.