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In solving money problems you have to keep track of the counts for the different coins as well as their values.
n = number of nickels
5n = value of the nickels in cents
d = number of dimes
10d = value of the dimes in cents
q = number of quarters
25q = value of quarters in cents
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$3.20 = 320 cents = 5n + 10d + 25q
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It is important to have everything in cents to avoid mismatched dollars and cents.
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q = 3n :: There are 3 times as many quarters as nickels, so you would have multiply n by 3 to be equal.
d = n+5 :: There are 5 more dimes than nickels.
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So, we can substitute what we know into the value equation.
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5n + 10d + 25q = 320
5n + 10(n+5) + 25(3n) = 320
5n + 10n + 50 + 75n = 320
90n = 270
n = 3
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Substituting n=3 we can find d and q.
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d = n+5 = 3+5 = 8
q = 3n = 3*3 = 9
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Check the work by making sure the value of these coin counts is $3.20.
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5n = 5(3) = 15 = $0.15
10d = 10(8) = 80 = $0.80
25q = 25(9) = 225 = $2.25
$0.15+$0.80+$2.25 = $3.20
Correct!
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Answer:
nickels = 3
dimes = 8
quarters = 9
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Done.

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The notion of '2x' is simply 2 times x, but I doubt that is what you mean...
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Consider the two linear equations as a system of simultaneous equations.
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Usually, you are looking for their point of intersection, or at least if they intersect or not.
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x + 2y =4
2x -y = 3
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Multiply the second equation by 2 to be able to eliminate the 'y' terms.
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2(2x-y)=2(3)
4x - 2y = 6
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Substitute this equation for the second equation and add them.
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x + 2y =4
4x - 2y = 6
5x = 10
x = 2
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Substitute x=2 into the the first equation to find y.
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x + 2y =4
2 + 2y =4
2y = 2
y = 1
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Now we have defined the point (2,1).
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We can substitute both values into the second equation to see if they are correct.
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2x -y = 3
2*2 -1 = 3
Correct.
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And you can graph it to see that they cross at (2,1). For this step having both equations in slope-intercept form is best for picking points to draw the lines: y = mx+b
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x + 2y =4
2y = -x + 4
y = (-1/2)x + 2
m = slope = -1/2
b = y -intercept = 2, which is the point (0,2)
x -intercept is where y=0, so we solve: 0 = (-1/2)x + 2, so x=4, which is the point (4,0)
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2x -y = 3
-y = -2x + 3
Multiply both sides by -1.
y = 2x - 3
m = slope = 2
b = y-intercept = -3, which is the point (0,-3)
x-intercept = 0 = 2x - 3, x = 3/2, which is the point (3/2,0)
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Note that we can tell the two lines are perpendicular because their slopes are negative reciprocals.
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The z distribution is defined as normally distributed with mean = 0 and s.d. = 1, shown as N(0,1).
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The mean is the "middle" or "center of mass" of the distribution, so 50% of the scores are less than z=0 and 50% are greater than z=0.

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-4x + 5y = 17
5y = 4x + 17
y = (4/5)x + 17/5
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therefore, the slope of this line is 4/5.
which means the slope of a perpendicular line is -5/4.
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We want this line with slope = -5/4 to go through the point (-2,5).
y = mx+b
5 = (-5/4)(-2) + b
5 = 10/4 + b
5 - 10/4 = b
5 - 2.5 = b
b = 2.5
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y = -5/4x + 2.5
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You can check with a graph. The red line is y = (4/5)x + 17/5. The green line is
y = -5/4x + 2.5
.

.

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Rearrange until you get it into slope-intercept form: y = mx+b.
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(-2/3)x+5=x+y-3
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Multiply everything by 3...
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3((-2/3)x+5)=3(x+y-3)
-2x + 15 = 3x + 3y - 9
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Subtract 3x from both sides...
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-5x + 15 = 3y -9
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Add 9 to both sides...
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-5x + 24 = 3y
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Divide both sides by 3...
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(-5/3)x + 8 = y
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y = (-5/3)x + 8
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slope = -5/3
y-intercept occurs where x=0, which is...
y = (-5/3)(0) + 8 = 8
or the point (0,8).
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You only need two points to draw a straight line, so the x-intercept is useful.
It occurs where y=0.
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0 = (-5/3)x + 8
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Multiply by 3
0 = -5x + 24
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Add 5x to both sides
5x = 24
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Divide by 5
.
x = 24/5 = 4+4/5 = 4.80
which is the point (4.8,0)
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You can check the graph to see...
.

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EQ.1.
2x+3y=9
3y = -2x + 9
y = -2/3x+3
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EQ.2.
3x-2y=8
-2y = -3x+8
y = 3/2x+4
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Since slope of EQ.1 is -2/3, a perpendicular line will have a negative reciprocal slope, which is 3/2.
EQ.2 has that slope, so they are perpendicular.
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1 hr = 3600 seconds
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26 miles/hr = 26 miles/1 hr = 26 miles/3600 seconds = .0072222 miles/second
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.82 seconds * .0072222 miles/sec = .0059222 miles
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That is not a very help measure, but we can multiply by 5280 ft/mile to get a better answer.
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5280 ft/mile * .059222 miles = 31.2693 ft
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As a quick reasonableness check, in school I memorized that 60 mi/hr = 88 ft/sec.
So in round numbers, 26 mph is about 30 mph = about 44 ft/sec.
And .82 is about .75 or 3/4 of a second.
3/4 of 44 = 33
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Answer:
The car travels about 31.2693 ft in .82 seconds at 26 mph.
.
Done

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x = first number
y = second number
x = 8y :: the first number = 8 times the second number
2y = x+1 :: twice the second number is one more than the first number
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substitute x = 8y...
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2y = 8y + 1
-6y = 1
y = 1/(-6) = -1/6
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x = 8y
x = 8(-1/6) = -8/6 = -4/3
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Checking...is twice the second number = 1 more than first?
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2*y = 2(-1/6) = -2/6 = -1/3
x+1 = -4/3 + 1 = -1/3
Correct
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Answer:
The numbers are x and y:
x = -4/3, and
y = -1/6.
.
Done.

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y = 30x +8000
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This equation is in slope-intercept form: y = mx+b. So,
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m = slope = 30, which is a very steep slope indeed
b = y-intercept = 8000, which is the point (0, 8000)
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x-intercept is where y=0
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0 = 30x + 8000
x = -8000/30 = -800/3 = -266 2/3, which is the point (-266 2/3, 0).
.

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factor
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Check using FOIL (first*first + outside*outside + inside*inside + last*last)...
.

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Done

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d = Derek's age
s = Sarah's age
d = 4s :: Derek is 4 times as old as Sarah
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3 years ago...
Derek's age = d-3
Sarah's age = s-3
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The sum of their ages was 44...
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(d-3)+(s-3) = 44
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Substitute d = 4s...
.
(4s-3) + (s-3) = 44
.
And there you go...

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D = R*T, where D=distance, R=rate of speed, and T=time
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S = speed in calm water = 15
C = current speed
S-C = speed going upstream
S+C = speed going downstream
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T = D/R
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Upstream...
T = 9/(S-C)
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Downstream...
T = 12/S
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Transitive property lets us eliminate T...which is good since we don't know what T is...
9/(S-C) = 12/S
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Cross multiply...
9S = 12*(S-C) = 12S - 12C
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Substitute S=15...
9(15) = 12(15) - 12C
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135 = 180 - 15C
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Add 15C to both sides...
15C + 135 = 180
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Subtract 135 from both sides...
15C = 45
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Divide both sides by 15
C = 3 = current speed
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Checking the answer by checking the time up and downstream..
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T = 9/(S-C) = 9/9 = 1 hr
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T = 15/(S+C) = 15/(12+3) = 15/15 = 1 hr
.
Correct.
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Answer:
The current is 3 mph.

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P = perimeter = 2L + 2W = 2(L+W)
P = 50
so...
L+W = 25
.
A = area = L*W
A = 150
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That means we know:
L + W = 25
and
L*W =150
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Substituting L = 25-W,
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(25-W)*W = 150
25W -W^2 = 150
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Add W^2 to both sides
.
25W = W^2 + 150
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Subtract 25W from both sides
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0 = W^2 - 25W + 150
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Factoring
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(W-10)(W-15) = 0
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That means W can be either 10 or 15.
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Since L = 25-W, L can be either 15 or 10, depending on W.
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Checking our work
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Is P = 50 if W=15?
2L + 2W = 2(10)+2(15) = 20+30= 50
Is P = 50 if L=15?
2L + 2W = 2(15)+2(10) = 50
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Either way the area is 150 because we know 10*15 = 15*10 = 150.
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Answer:
L = 10 and W = 15
OR
L = 15 and W = 10
.
Done

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There is no 'point-intercept', so you can forget that.
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y = mx+b is the slope-intercept form. You can tell because you are given the slope, 'm', and the y-intercept, 'b'.
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The other form is the point-slope form. You can tell this because you are not given the intercept or, by the process of exclusion, you know it is not the slope-intercept.
The point-slope form is
y-y₁ = m(x-x₁)
.

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A trend line shows the average of a series of points over time.
We can assume the independent variable, age, is the x-axis.
That means height is a function of age, that is f(x), so it will be plotted on the y-axis.
The data points you have are:
(0,4)
(4,7)
(7,11.5)
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To draw a line, you need two points.
(0,4) should be one of them.
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The slope-intercept form is: y = mx+b
In this case, we know that when x=0, y=4, so the y-intercept has to be 4.
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The slope is the rise/run, which is the change in y divided by the change in x.
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From (0,4) to (4,7),
change in y = 7-4 = 3
change in x = 4-0 = 4
m = 3/4 = .75
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From (4,7) to (7,11.5),
change in y = 11.5-7 = 4.5
change in x = 7-3 = 4
slope = 4.5/4 = 1.25
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So the slope is getting steeper, which means a straight line will have to miss one or both of the points. That means the trend line will be somewhere between them.
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We can pick the average value of 'x' and 'y' as the point that will be the second point on which we will base our trend line.
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The average value of 'y' = (7+11.5)/2 = 18.5/2 = 9.25
The average value of 'x' = (4+7)/2 = 11/2 = 5.5
So the point is (5.5,9.25)
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Based on the point (0,4) as our starting point...which reasonable since that is the at birth height...
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change in y = 9.25-4 = 5.25
change in x = 5.5-0 = 5.5
slope = 5.25/5.5 = .954545
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Recall our second point will be (5.5,9.25)
The equation will be:
y = .954545*x + 4
.

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1 cubic foot of water is about 7.48 gallons.
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The volume = Area of the circle * height of the column. This is the formula for the volume of a cylinder, which is what you will have.
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Area = pi*r^2 = pi*7.5^2 = 176.71 square feet
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Assuming you want to fill the pool to the rim, that is 42 inches, which = 3.5 ft.
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3.5 ft * 176.71 ft^2 = 618.485 ft^3
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618.485 ft^3 * 7.48 gallons/ft^3 = 4626.27 gallons
.
Done

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The basic distance equation is useful: D = RT, where D=Distance, R=Rate of speed & T = Time.
D_C = Chuck's distance = 80 km
D_M = Mary's distance = 180 km
T = Chuck's time = Mary's time
R_C = Chuck's rate of speed
R_M = Mary's rate of speed
R_M = R_C + 50 kph :: Mary is going 50 km/hr faster.
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Since they travel for the same time, but you do not know the time, you need to eliminate it.
.
We know the distance travelled at speed = R_C.
.
80 = R_C*T
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So we can define T = 80/R_C
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And we know the distance travelled at speed = R_M.
.
180 = R_M*T
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So we can define T = 180/R_M
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Using the transitive property to eliminate T.
.
80/R_C = 180/R_M
.
Substitute R_M = R_C+50.
.
80/R_C = 180/(R_C+50)
.
Cross multiply to eliminate denominators.
.
80*(R_C+50) = 180*R_C
.
80*R_C + 4000 = 180*R_C
.
4000 = 180*R_C - 100*R_C
.
4000 = 100*R_C
.
R_C = 4000/100 = 40 km/hr
.
Now we can substitute that value to get R_M.
.
R_M = R_C+50
.
R_M = 40 + 50 = 90 km/hr
.
Checking to see if these are the answer.
.
80 = 40*T
T = 80/40 = 2 hr
.
180 = 90*T
T = 180/90 = 2 hr
.
Correct!
.
Answer:
Chuck's speed is 40 km/hr.
Mary's speed is 90 km/hr.
.
Done.

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y = -.00635*x^2+630

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Checking...
.
??

Correct.
.
Answer: a=4
.
Done

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3x -2/x + 7 = 0
.
Multiply by x to remove denominator.
.
x(3x -2/x + 7) = x(0)
3x^2 -2 +7x = 0
.

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 : . Discriminant d=73 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 0.257333957552922, -2.59066729088625. Here's your graph:

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When you know two points, use the slope-intercept form: y = mx+b
m = rise / run = change in y / change in x
.
m = (4-3) / (0-3) = 1/-3 = -(1/3)
.
y = -1/3x+b
.
Now substitute one of the point's values to find b.
.
4 = -(1/3)(0) + b
b = 4
.
Check using the second point.
.
3 = -1/3(3) + b = -1 + b
b = 4
.
Therefore,
y = -1/3x + 4
.
Graph it for another check.
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Is y= -2x -4 a linear equation?
Yes, it is.
Since there is no exponent, there will be no curves.
So it is a straight line, which means it is a linear equation;.
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D = RT, which looks like the standard distance equation, where
D = Distance travelled
R = Rate of speed
T = Time
.
D = RT
.
To find R, divide both sides by T
.
D/T = RT/T = R*(T/T) = R*1 = R
.
D/T = R
R = D/T
.
Done

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The 'addition' method for solving systems of linear equations is also known as the 'elimination' method.
.
First, get the equations into standard form: Ax + By = C
.
5x -40 = 6y
5x - 6y = 40 :: (EQ 1)
.
2y = 8 - 3x
3x + 2y = 8 :: (EQ 2)
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If we multiply EQ 2 by 3, the two equations will be in shape to eliminate the 'y' term.
.
5x -6y = 40 :: (EQ 1)
9x + 6y = 24 :: (EQ 2 * 3)
.
adding...
.
14x = 64
.
x = 64/14 = 32/7
.
substitute x = 32/7 in EQ 1 to find y
.
5(32/7) -6y = 40
160/7 -6y = 40
.
multiply both sides by 7 to remove the denominator
.
160 -42y = 280
.
subtract 120 from both sides
.
-42y = 120
.
divide both sides by -42
.
y = -120/42 = -60/21
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So, we think the intersection of the two lines will be: (32/7, -60/21) = (4 4/7, -2 6/7)
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Graphing is good way to check.
The slope-intercept form is a good way to compute the lines: y = mx+b.
.
5x-40=6y
.
6y = 5x -40
.
divide by 6
.
y = 5/6x - 40/6
y = 5/6x - 20/3
.
2y=8-3x
2y = -3x + 8
y = -3/2x + 4
.

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An average of 88% means that the sum of the four tests divided by 4 = .88 = 88%.
S = sum of the 4 scores
S/4 = 88
S = 4*88
S = 352
.
We need to find the value of the 'next' test that will bring the average to 90%.
.
(352+x)/5 = 90
.
Multiply both sides by 5 to eliminate the denominator
352 + x = 450
x = 450 - 352 = 98
.
So, the student needs to score 98% on the fifth test to pull the average up to 90%.
.
Check by computing the average...
352 + 98 = 450
450 / 5 = 90
.
Answer: The student needs to score 98% on the fifth test to bring the average up to 90%.
.
Done.

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y = mx + b is the basic slope-intercept form of a linear equation, where
m = slope
b = y-intercept (recall, the y-intercept is the point where x=0)
.
slope = -3 = m
.
y-intercept = 5 = b,
or
(0,5)
.
y = -3x + 5
.
A good check is what is the x-intercept, and then graph it.
.
The x-intercept is where y=0.
.
0 = -3x + 5
.
x=5/3
.
or (5/3,0)
.
Graphing can be used to check
.

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The vent fills the pool in 10 hours, so it fills at a rate of 1/10 of the job per hr.
The drain empties the pool in 20 hours , so it drains at a rate of -1/20 per hr.
We can imagine the pool being full as = 1.
.
x(1/10) -x(1/20) = 1 job
.
Multiply through by 20.
.
2x - x = 20
x = 20
.
Checking, we can find how much water was put in vs. how much water was pulled out.
.
20 hr * 1/10 per hr = 2 :: So in 20 hrs the pool can be filled twice.
20 hr * 1/20 per hr = 1 :: So in 20 hrs the pool can be drained once.
2-1 = 1, which suggests the pool is full at that instant.
.
But I suspect the pool cannot hold twice it's normal volume. That is, once you have it full, adding more water just flows over the rim and floods the building. So, perhaps 20 hrs is too long.
.
Perhaps a better model is to suggest the pool is filling and draining simultaneously. That means it fills at a net rate of 1/10 - 1/20 per hr.
.
x(1/10-1/20) = 1
x(1/20) = 1
x = 20
.
Hmmm...20 again.
.
So, the answer is 20 hrs.
.
But, after 20 hrs the pool will be flooding the building since it will still be adding water at the rate of 1/10 the volume of the pool per hr minus 1/20 of the volume per hr or a net rate of 1/20 of the volume per hr.
.
Done.

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.

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Since

We can rearrange to say that
,
which suggests you divide A/W (that is, polynomial division).
.
But you may be able to factor the equation to advantage.
.

.
So substituting what we know
.


.
Which means

.
Done

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A number = x.
4 is added to twice a number = 2x + 4
4 is added to three times a number = 3x + 4
2x + 4 = y
3x + 4 = y
.
By the transitive property...
.
2x + 4 = 3x + 4
.
0 = x
.
x = 0
.
So the answer is x = 0.
.
Any number times 0 = 0, so that's logically the only possible answer.
.
Done

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The four points can be plotted:
.

.
The diagonals are defined by the points: U,W and V,X.
.
We need to solve for a line going through the first pair of points
U (-2,1)
W (4,-1)
.
y = mx + b
m= (-1-1)/(4-(-2))= -2/6 = -1/3
y = -1/3x + b
We can solve for 'b' by plugging in the points we know.
1 = -1/3*(-2 ) + b
b = 1/3
or
-1 = -1/3*4 +b
b = 1/3
.
y=-1/3x + 1/3 defines the equation for the first diagonal.
.
Then we need to find the equation for the line going through the other two points
V (2,3)
X (0,-3)
m= (-3-3)/(0-2) = -6/-2 = 6/2 = 3
.
At this point we know the diagonals are perpendicular because, by definition, the slopes of perpendicular lines are the negative reciprocal.
-1/3 is the negative reciprocal of 3.
.
But we can press on to continue to define the second equation...
.
-3 = 3(0) + b
b = -3
or
3 = 3(2) + b
b = -3
y = 3x -3
.
Then a graph will illustrate they are perpendicular.
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