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We have 10 batteries, so let's say the number of 6V batteries is x. That means the number of 12V batteries is 10 - x.
And we're told that 6x + 12(10-x) = 84 volts. 6x is the 6 volts per battery times x, which is the number of 6V batteries. Likewise 12(10-x) is the 12 volts per battery times the number of 12V batteries.
So, we have 6x + 120 - 12x = 84. Combining the x terms we have -6x + 120 = 84. Subtracting 120 from both sides, we have -6x = -36. Dividing by -6, we have x = 6.
That means the number of 6V batteries is 6, and the number of 12V batteries is 10-6=4.
ALWAYS check! 6*6=36 and 4*12=48, 36+48=84. Check.

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Squaring both sides, we have 4x + 1 + 3 = 0. Simplifying, 4x = -4, which means x = -1.

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1) Problems like this build on the fact that something has to equal something else. In this case, we know they both traveled for 4 hours. The distance equation is d = r * r, so for Car1 d1 = r*4 and for Car2 d2 = (r+15)*4. And we know that d1 + d2 = 460 miles (the distance they are apart after 4 hours). Substituting, we have 4r + 4r + 60 = 460. That simplifies to 8r = 400, so r = 50. That means r+15 is 65. Car1 traveling at 50 and Car2 is traveling at 65.
ALWAYS check your answer. 4*50 = 200 miles. 4*65 = 260. 200+260=460. Check.
2) Total Revenue = $12 times the number of shirts sold (call that x). Total Cost is $20 plus $8 * x. Profit is total revenue minus total cost. So, we now have some linear equations to solve, with only one variable.
, which is the total revenue.
, which is the total cost.
, which is the desired profit.
, which is the standard profit equation.
Therefore, by substituting what we know:
.
Simplifying, we have:
, so .
ALWAYS check your answer! So TR = 30*$12 = $360; TC = 20 + $8*30 = $260; which means P = 360 - 260 = 100. Check.

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When the stray elephant caught it, it had traveled the same distance as the herd. Otherwise, it would not have caught up with them. (Right?)
Therefore, the herd had to have walked .83 of a mile. At 6 miles per hr, the herd covers 1 mile every 10 minutes. So they had walked for 8.3 minutes at 6 miles per hour to cover .83 miles.
Think of the basic equation: d = r * t. Distance = Rate * Time. We know the herd traveled .83 miles and that they traveled at 6 miles/hr. So we divide .83 by 6 and get .138333 hours. Times 60 min/hr, we get 8.3 minutes.
With a problem like this one, it is a good idea to get everything into the right units of measure. Then looking at the units of measure you need in your answer should help you decide what to do next.
So, the equation for the distance traveled would be: d = 6 mi/hr * .138333 hr.

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The mean is the sum of the values divided by the count of the number of values. The sum of the values is 4.8 + 1.6 + 5.2 + x. The number of values equals 4. Thus we know that

Multiplying by 4, we have

Subtracting 13.6 from both sides, we have the answer

We now check our answer by substituting the value of x into the original equation and solve it, to make sure the mean is 3.7.

. Check.

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Substituting for y, 10x + 2(-5x + 3) = 7. That simplifies to 10x - 10x + 6 = 7, or 6 = 7. That is not possible, so there is no solution for the simultaneous equations. One or the other of the equations is false.

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It takes .6 sec to fill a bottle and move the next one into place to be filled. In one hour (3600 sec), the machine can fill 6000 bottles (that is 3600/.6). In two hours, it can fill 12000.

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Parallel lines have the same slope (by definition), so the slope of the line passing through the point(-2,1) will have to have slope = -2/3. We know that the line is sloping down to the right because the slope is negative. The slope is also the change in y divided by the change in x, so for each change of 3 on the x-axis, the y value decreases by 2 (that is, -2/3). To define a line, we need another point to pair with (-2,1).
We know that the general equation for a line is . We know one y and one x, but we don't know b, so we substitute what we know and solve that.

so
Recall the b is called the y-intercept, which tells us where the line crosses the y-axis when x=0. That gives us another coordinate pair: (0, -1/3). So the equation for the line is:
.

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Fill the 3c container.
Pour it into the 5c container.
Refill the 3c container.
Pour into the 5c container until the 5c container is full.
You will have 1c left in the 3c container.

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You've left something out.
sqrt of 4 = 2, and 2+1+3 cannot equal 0.
sqrt or (4+1+3) = sqrt of 8 = 2 * sqrt 2, which is not equal 0.

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The statement that the "length is 1 cm longer than the width" means that we can define the length (l) = w + 1. Width (w) is simply = to w. So, the rectangle has sides of w, w+1, w, and w+1. The diagonal is known to be 4 cm (given).
Now we simply apply the Pythagorean formula. Let's call the diagonal d. That means


Therefore,
Of course, is the square of the diagonal, and we were told the diagonal was 4 cm, so .
Substituting what we know:

Subtracting 16 from both sides:

This does not factor easily, so you can use the quadratic equation to find w.
Quadratic formula:
Substituing, we have

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Using brackets helps visually, but nested parentheses are okay, too.
4[2(-5x+y)-y]-10(y-4x) means 4 * (2 * (-5x + y) - y) - 10*(y - 4x).
Then you simplify the innermost parentheses repeatedly. But you have to remember the rules of precedence: exponentiation, then multiplication and division (left to right), then addition and subtraction (left to right).

4 * (2 * (-5x + y) - y) - 10*(y - 4x) becomes
4 * (-10x + 2y - y) - 10*(y - 4x) because you do the multiplication before subtracting the y.
Solving inside the parentheses is next:
4 * (-10x + y) - 10*(y - 4x)
Now multiply left to right:
-40x + 4y - 10y + 40x because mutiplying -4x by -10 is +40x.
The -40x cancels + 40x, so we end up with -6y

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To compute volume you multiple length by width by depth (or height, as the case may be). BUT, all of the values must have the same units of measure.
Here you have 42" for one dimension. The other two dimensions are in feet: 6 feet and 3 feet.
So, you have to convert 42" to feet, or 6 feet and 3 feet to inches, before you multiply.
Dividing 42" by 12" per foot, you get 3.5 feet. So the volume is simply 3.5 times 3 times 6 cubic feet.

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On the first day Jesse swam fewer meters than the second day, so let's call that x. The total is 2050, so on the second day Jesse swam 2050-x meters.
We also know that on the second day Jesse swam 250 meters more than the first day.
Recall that the first day was x and that the second day is 2050-x. So to make them equal, you have to add 250 meters to the first day, so it becomes x+250.
x + 250 = 2050 - x
Adding an x to both sides of the equation, we obtain:
2x + 250 = 2050
Subtracting 250 from both sides, we obtain:
2x = 1800
Dividing both sides of the equation by 2:
x = 900
So, Jesse swam 900 meters on the first day.
Adding 250 we get 1150, which is the number of meters swam on the second day.
Checking these answers, we find that 900 + 1150 = 2050, which is the correct total.

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The solution is found by applying the choices given.
a. By inspection, you can see that the variable y can be eliminated by adding the equations, provided you multiply the first equation by 3 and the second equation by 4. (We're actually getting at the least common multiple of 4 and 3, which is 12.)
3(5x - 4y = 6) becomes 15x - 12y = 18
4(2x + 3y = 2) becomes 8x + 12y = 8
Adding the equations, we obtain: 23x = 26, which is one equation with one variable. So, "a" is clearly the answer.
At this point, you are done. The other choices have to be wrong. However, if one of the choices were to be "All of the above" you have to test at least one more alternative.
b. This choice calls for multiplying the terms in the first equation by 2, the terms of the second equation by 5, which is a good start. But it unravels when it concludes "and then add the equations." To eliminate the x-terms, you would need to subtract the equations, not add.
c. This choice calls for multiplying the terms in the first equation by -3, the terms of the second equation by 4, and then adding the equations.
-3(5x - 4y = 6) becomes -15x + 12y = -18
4(2x + 3y = 2) becomes 8x + 12y = 8
When you add them, you do not eliminate a variable, so this choice is wrong.
d. Adding the equations does not eliminate a variable.
5x - 4y = 6
2x + 3y = 2
-----------
7x - y = 8

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The sigma means to sum the values, substituting successive integer values of k in the equation from k=1 to k=infinity. (Of course, k cannot equal infinity, so it just approaches it.)
When k=1, the entire denominator is raised to the zeroth power, so it equals 1. And 1/1=1. All other values of k will lead to the denominator being larger and larger, which means the resulting value of the division gets smaller and smaller.

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Substituting -4 for x in the equation, you have = = -14.

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Given: x - 2/4*x - 7/2*x + 1 = 0.
This simplifies to x - 1/2*x - 7/2*x + 1 = 0.
Or x - 8/2*x + 1 = 0.
Or x - 4x + 1 = 0.
Or -3x + 1 = 0.
That means -3x has to = -1 for the equation to = 0.
So x is simply 1/3.

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The general formula is y = mx + b.
Your given equation is 4x - y = 0.
Adding y to both side of the equation yields: 4x = y, which is the same as saying y = 4x.
In the general formula, the coefficient m is the slope.
So in this problem, the slope is the coefficient of x, which is 4.
Note that in the general formula, "b" is simply a constant that is the same as the y-axis intercept (or y-intercept). In this case, it is zero.

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The rules of precedence require you to first do the exponentiation, so you must first solve (a-b)^2.
Using the distributive property, (a-b)*(a-b) = a^2 - 2ab + b^2.
Then you substitute into your original equation:
4(a^2 - 2ab + b^2) + 7(a + b).
That gives you 4a^2 - 8ab + 4b^2 + 7a + 7b.

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The simple rule for computing a percentage change when given an original number (oldValue) and a new number (newValue) is:

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You're close. Try: (5t + 1)(3t - 4).
Using the distributive property, you will obtain:
15t^2 - 20t + 3t - 4, which simplifies to 15t^2 - 17t - 4.
Then solve for t by setting each factor to zero. That is,
(5t + 1) = 0 and (36t - 4) = 0.
For 5t + 1 to equal zero, then 5t must = -1 (i.e., -1 + 1 = 0), so t = -1/5.
For 3t - 4 to equal zero, then 3t must equal 4 (i.e., 4 - 4 = 0), so t = 4/3.
Therefore, your solution set might be t = { -1/5, 4/3 }
I say "might be" because there may be a constraint.
If t cannot be a negative number, then your only solution is 4/3.
Likewise, if t cannot be a positive number, then your only solution is -1/5.

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Given: 2w(4w+1)=1
We know that 2*w*(4*w+1)=1 (which simply shows the computations).
Using the distributive property: 2w(4w+1)=8w^2+2w
So 2w(4w+1)=1 becomes 8w^2+2w=1 or
8*w^2+2*2 (showing the computations).
Subtracting 1 from both sides of the equation: 8w^2+2w-1=0, which is standard form.
We cannot factor 8w^2+2w-1 into simpler terms, so we have to use the quadratic equation: .

Consequently, .

This simplifies to .

So x = .25 or -.5.