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y = David's age
.
Tom is 2 years younger than David.
.
y-2 = Tom's age

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A 15% increase means the schools' current enrollment = 115% of last year's enrollment.
x = last year's enrollment
115% * x = 552
1.15*x = 552
x = 552/1.15
x = 480

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Three consecutive odd integers can be defined as:
x
x+2
x+4
.
The sum of these three number is 36 less than the product of the smaller two.
.
x +x+2 +x+4 +36 = x*(x+2)
.
Solve
.
3x + 42 = x^2 + 2x
.
x^2 -x -42 = 0
.
Factor
.
(x-7)(x+6) = 0
.
x = 7 or -6
.
-6 is not odd, so we select 7 as the answer.
.
x = 7
x+2 = 9
x+4 = 11
.
Check the values.
7*9 = 63
7+9+11 = 27
27+36 = 63
Correct
.
Answer: The numbers are 7, 9, and 11.
.
Done.

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The two cars are moving in opposite directions, so they are separating at the sum of their speeds.
Looking at the fundamental distance equation, d = rt, we know:
d = 425 km
t = 5 hr
So, the combined rate = 425/5 = 85 km/hr
One car goes 5 km/hr faster than the other, which means we can define their speeds as x and x+5.
x + x+5 = 85
2x +5 = 85
2x = 80
x = 40, which is the speed of one car
x+5 = 45, which is the speed of the other car

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2x + 9y = 0
3x + 5y = 17
.
multiply the first equation by 3 and the second by 2; then subtract
.
6x + 27y = 0
6x + 10y = 34
--------------
17y = -34
.
y = -2
.
substitute to find 'x'
.
2x + 9(-2) = 0
2x -18 = 0
2x = 18
x = 9
.
check by substituting these values into the other equation
.
3(9) + 5(-2) = 17 ??
27 -10 = 17
correct
.
Answer: (9,-2)
.
Done.

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5 cent coins are called nickels.
10 cent coins are called dimes.
.
8 nickels = 8 * 5 cents = 40 cents
3 dimes = 3 * 10 cents = 30 cents
40 cents + 30 cents = 70 cents
.
Done.

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2x + 9y = 0
3x + 5y = 17
.
multiply the first equation by 3 and the second by 2; then subtract
.
6x + 27y = 0
6x + 10y = 34
--------------
17y = -34
.
y = -2
.
substitute to find 'x'
.
2x + 9(-2) = 0
2x -18 = 0
2x = 18
x = 9
.
check by substituting these values into the other equation
.
3(9) + 5(-2) = 17 ??
27 -10 = 17
correct
.
Answer: (9,-2)
.
Done.

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The consecutive angles of a parallelogram are supplementary.
Call these angles x and y.
x + y = 180
x = 2y+6
substitute
2y+6 +y = 180
3y = 174
y = 58
x = 2*58 + 6 = 122
Done.

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The probability of picking a girl on the first pick is: 49/100.
.
The probability of picking a girl on the second pick depends on whether a girl was picked first.
If a girl was picked initially, then the probability of picking a girl on the second pick is : 48/99.
If a boy was picked initially, then the probability of picking a girl on the second pick is: 49/99.
.
The probability of picking a girl first and second is: 49/100 * 48/99.
.
And so forth.
.
You need to determine the conditional probabilities to find the chance of picking at least 1 girl.
.

1st Pick	2nd Pick	3rd Pick	Probability	Description
P(girl)=49/100	P(girl)=48/99	P(girl)=47/98	P=0.113939394	(A) 3 girls
"	"	P(boy)=51/98	P=0.123636364	(B) 2 girls, 1 boy
"	P(boy)=51/99	P(girl)=48/98	P=0.123636364	(C) 2 girls, 1 boy
"	"	P(boy)=50/98	P=0.128787879	(D) 1 girl, 2 boys
P(boy)=51/100	P(girl)=49/99	P(girl)=48/98	P=0.123636364	(E) 2 girls, 1 boy
"	"	P(boy)=50/98	P=0.128787879	(F) 1 girl, 2 boys
"	P(boy)=50/99	P(girl)=49/98	P=0.128787879	(G) 1 girl, 2 boys
"	"	P(boy)=49/98	P=0.128787879	(H) 3 boys

The total probability = 1.00, as you can see.
.
P(at least 1 girl) = A+B+C+D+E+F+G = 0.871212121
.
OR
.
You could see the problem in terms of the chance of picking 3 boys. If you do not pick 3 boys, then you have to pick at least 1 girl.
.
P(picking 3 boys) = 51/100 * 50/99 * 49/98 = 0.128787879
.
P(picking at least 1 girl) = 1.0 - 0.128787879 = 0.871212121
.
Same answer. Two ways to get there.
.
Done.

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5) The number in the hundredths place is 4.
so the number is:
xy.z4
.
4) The number in the tens and hundreths place are the same.
.
4y.z4
.
3) The number in the ones and tenths are the same.
.
Of course they can be 2's or 4's.
So, the number can be either
42.24
or
44.44
.
2) The numbers can be only 2's or 4's.
OK.
,
1) The number has 4 digits.
OK.
.
So the number can be 42.24 or 44.44.
.
Done.

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f(x) = 7*x
.
x = .75
.
f(x) = 7*.75 = 5.25

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x = number of students 25 and under
y = number of students over 25
.
x + y = 8500
so
x = 8500 -y
.
y = 1/3x
so
x = 3y
.
substitute
.
x = 3(8500-x)
.
x = 25500 -3x
.
4x = 25500
.
x = 6375
.
8500-x = 2125
.
The number of students 25 and under is 6375.
.
The number of students over 25 is 2125.
.
The number of students over 25 = 2125/6375 = 1/3 the number of students 25 and under.
.
Done.

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6x+y = 4
y = -6x + 4
.
This equation defines a line.
.
The y-intercept is (0,4)
.
To find the x-intercept, set y=0 and solve for 'x'.
.
0 = -6x + 4
6x = 4
x = 2/3
which is the point (2/3,0)
.
With two points you can draw a line.
.

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Get both equations into slope-intercept form and determine if their slopes are inverse reciprocals.
.
x +2y = -10
2y = -x -10
y = -1/2*x -5
.
-4y = 2x + 20
y = -2/4*x + 20/-4
y = -1/2*x - 5
.
The lines' slopes are equal, so the lines are parallel.
In fact, the lines lie one on top of the other such that they appear to be same line.
.
Answer: False. The lines are not perpendicular.
.
The graph is useful.
.

.
Done.

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m = -4, so the basic equation will be
y = -4x + b
.
We're given the y-intercept, which is the point (0,b).
.
y = -4x -3/5
.
A graph is useful.
.

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With coin problems you have to keep track of the count of each coin and the value.
n = number of nickels
5n = value of the nickels in cents
d = number of dimes
10d = value of the times in cents
.
5n + 10d = 765 cents
.
2(10d) + 5(n+35) = 1430 cents
20d + 5n + 165 = 1430
20d + 5n = 1265
.
With two equations and two unknowns, you can solve the system of equations.
.
5n + 10d = 765
5n + 20d = 1265
----------------
-10d = -500
.
d = 50
.
Substitute d = 50 to find n.
.
5n + 10d = 765
5n + 10(50) = 765
5n = 265
n = 53
.
Check the values to make sure the above solution is the correct answer.
.
5(53) = 265
10(50) = 500
265 + 500 = 765 cents
.
Check the other equation, too.
.
2(10(50)) + 5(53+35) = 1430 ??
2(500) + 265 + 175 = 1430
Correct.
.
Answer: Joe has 53 nickels and 50 dimes.
.
Done.

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3x=-3x+12
.
Add 3x to both sides.
.
6x = 12
.
Divide both sides by 6.
.
x = 2

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I assume the trains are heading toward each other from the two different stations.
That means they are converging at a combined speed of 90 mph.
When they meet, they will have covered the 360 miles.
The fundamental distance equation is d = rt.
In this case we know d = 360 miles and r = 40+50 = 90 mph. Therefore,
t = 360/90 = 4 hr
The train going 40 mph will have covered 4*40 = 160 miles at the time they meet.
The train going 50 mph will have traveled 4*50 = 200 miles at the time they meet.
160+200 = 360 miles, so that checks.
Re-reading the question, it asks for how far from the nearest station they will be.
Well, both of them will be 160 miles from the nearest station. One train will heading toward the nearest station and the other will be heading away from it.
Done.

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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 : . The discriminant -20 is less than zero. That means that there are no solutions among real numbers. If you are a student of advanced school algebra and are aware about imaginary numbers, read on. In the field of imaginary numbers, the square root of -20 is + or - . The solution is Here's your graph:

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7x + 12 = 9
.
subtract 12 from both sides
.
7x = -3
.
divide both sides by 7
.
x = -3/7
.
Done.

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y = 2x-4
.
A line perpendicular to this line will have a slope that is the inverse reciprocal of 2, which is -1/2.
.
y = -1/2x + b
.
To determine 'b', use the point you want to go through.
.
3 = -1/2(-1) + b
3 = 1/2 + b
b = 2 1/2 = 5/2
.
A graph usually helps. The red line is y = 2x-4; the green line is y = -1/2x + 5/2.
.

.

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Two consecutive positive integers can be defined as 'x' and 'y', but then you have two unknowns.
So, the better way to define to consecutive integers is as 'x' and 'x+1'. Then it will go smoothly.
.
x + (x+1) = 650 would be the sum, not the product.
.
The product is defined by:
.
x*(x+1) = 650
.
Solving this...
.
x^2 +x -650 = 0
.
To factor, we need to find two multiplicands that a 1 apart and total 650.
A good place to start is the square root of 650.
Using the Google built-in calculator, just type 'sqrt(650)' in the Google search box to find:
sqrt(650) = 25.4950976
.
Testing 25 and 26, we find their product is 650. Therefore, the quadratic can be factored as:
.
(x+26)(x-25) = 0
.
That means x = -26 or +25.
.
We're looking for positive integers, so x=25 is chosen.
x+1 = 25 + 1 = 26
.
Answer: Two consecutive positive integers whose product is 650 are 25 and 26.
.
Done.

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x is the unknown number
.
5/6*x = -60
.
multiply both sides by 6
.
5x = -360
.
divide both sides by 5
.
x = -72

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The ratio of 5:4 means the 90 degrees of the complementary angles are divided into 9 parts.
90/9 = 10 degrees.
So, one angle is 5*10 = 50 degrees; and the other is 4*10 = 40 degrees.
50/40 = 5/4, which checks.
Answer: The smaller angle is 40 degrees.

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y = 1/3x -3
.
x = 0, y = -3
.
y = 0, x = 9
.
With two points you can draw a line.
.

.
The point (-3,4) is not on the line.
.
You could construct a perpendicular line that goes through (-3,4), but it is not clear what you need.
.
A perpendicular line will have the inverse reciprocal slope, which is -3.
.
y = -3x + b
.
Use the point to fine 'b'.
.
4 = -3(-33) + b
4 = 9 + b
b = -5
.
y = -3x -5
.

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Assume there are 52 cards in the deck.
13 of these are diamonds.
.
There are 13 favorable outcomes.
There are 52 possible outcomes.
.
13/52 = 1/4 = 25%
.
Done.

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For any given perimeter, the maximum area is a square.
.
5000/4 = 1250
.
1250^2 = 1562500 sq. ft.
.
If you're not familiar with the maximum area rectangle being a square, a simple application of calculus will demonstrate.
.
Recall, perimeter = 2(length + width). With 5000 feet of fencing, that is the maximum perimeter.
.
From this starting point, we recognize that 1/2 the perimeter equals the rectangle's length + its width.
.
5000/2 = 2500
so
L + W = 2500
.
So the sides can be defined as 'L = x' and 'W = 2500-x'.
.
y = area = L*W
.
Substitute for L and W.
.
y = x*(2500-x)
.
y = -x^2 + 2500x
.
To find the maximum, take the first derivative and solve it for a value of 0.
.
dy/dx = -2x + 2500
.
2x = 2500
.
x = 1250 = L
.
2500-x = 1250 = W
.
L = W, so we have a square.
.
Done.

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.

.

.
Of course, x can be + or -.

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The total interior degrees of polygon = (n-2)*180, n = number of sides.
Note that 'n' also define the number of interior angles.
Think about it: a triangle has 3 sides and 3 interior angles, a pentagon has 5 and 5, etc.
.
Divide the total interior angles by the number of angles, which will equal 162 degrees (as is given).
.
(n-2)*180 /n = 162
.
180n - 360 = 162n
.
18n = 360
.
n = 20 sides, which means there will 20 angles, too.
.
Check the answer.
.
(n-2)*180 = 18*180 = 3240 degrees
.
3240/20 =162 degrees
.
Done.

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(x,y) = (x,5)
.
(x₁,y₁) = (7,2)
.
m = (y - y₁) / (x - x₁)
.
m = (5-7) / (x-2)
.
m = -2 / (x-2)
.
Done.

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With 4 kings, you have 2 black kings and 2 red kings.
There are 2 ways you can draw a black king from the 4 possibilities.
2/4 = 1/2 = 50%