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Let x be the first part, then 2x is the second part and x+6 is the third part.
x + 2x + x+6 = 78
4x+6 = 78
4x = 72
x = 18
2x = 36
x+6 = 24
Check:
18+36+24 = 78

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Let the original side length be x.Then,
Solve for x.
Subtract 289 from both sides.
Solve this quadratic equation for x by factoring.

so or
so Discard the negative solution as the side length is a positive value.
The original side length is 12 cm.
Check:

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=
Note:
= Right? Notice that it is NOT (-4)(-4) which would be +16
= OK?
Why? Because anything raised to the zero power is equal to 1.

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Which strategy will work?
Which ones have you tried???
In fact, all of the listed strategies will work!
a) Let's do the graph and see how it works.
First, solve each equation for y.
1) 2x+y = 4 Subtract 2x from both sides.
1a) y = -2x+4 (Red line on the graph)
2) x-y = 2 Add y to both sides.
2a) x = y+2 Now subtract 2 from both sides.
x-2 = y or y = x-2 (Green line on the graph)
Now these can be graphed:

You can see that the two lines intersect at x = 2, and y = 0 so the solution is: (2, 0)
b) Solve the first equation for y:
2x+y = 4 Subtract 2x from both sides.
y = -2x+4 now substitute this into the 2nd equation:
x-y = 2 Substitute y = -2x+4
x-(-2x+4) = 2
x+2x-4 = 2 Add 4 to both sides.
3x = 6 Divide both sides by 3.
x = 2 Substitute this into the first eqation:
y = -2(2)+4
y = -4+4
y = 0
The solution is:
(2, 0)
Try c on your own and you will see that all three methods will work.

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You could with the conversion formula:
1) or:
2)
Since you want F = C you can substitute in either equation and solve.
1) Solve for F. Subtract from both sides.
Now multiply both sides by

2) Multiply both sides by 9.

Subtract 5C from both sides.
Divide both sides by 4.

The answer is at -40 degrees the temperature on the Farenheit scale reads the same on the Celsius scale.

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Factor completely using integer coefficients:
e) First, factor an x.
Now factor the parentheses. I know that my factors here will be two binomials of the form:
where and and , so m = -2 and n = 8. So the parentheses, when factored, look like:
, then the final answer is:

f) First, factor 2x.
Now factor the parentheses. I know my factors here will be two binomials of the form: where: and so and , so the parentheses, when factored, look like:
, so far we have:
but notice that is the difference of two cubes and this can be factored.
The difference of two cubes is factored thus:
so, in this case:
putting it all together, we have:

j) Factor as:
But again, we have the difference of two cubes:, so...
or:

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Simplify: You don't want to see the square root in the denominator.
Rewrite as:
Multiply top & bottom by to rationalise the denominator.
Simplify.

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Try this!
6 men in 14 days = x men in 21 days. Written as a proportion:
Multiply both sides by 21.
Simplify.


It'll take 9 men to do the same job in 21 days.
//////////////////////////////////////////////
My answer here is incorrect!
Please see the correct answer (#72623) of 4 men given by ankor.

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Let x be the smallest integer, then the next consecutive integer is x+1 and the next one after that is x+2. The sum of these is:
Simplify and solve for x.
Subtract 3 from both sides.
Divide both sides by 3.

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Let's first see how much you poured out:
litre has been removed.
Since you had liter to start with, you now have = litre remaining.

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Factor(ise) the following:
Here, you have the difference of two squares which can be factored thus:

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Try this:

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1) "In a right triangle, the length of the hypotenuse is 5 and the length of the legs is 3" ???
This cannot be true!
Remember that in any right triangle, the length of the hypotenuse squared is equal to the sum of the squares of the two legs. If the dimensions of the right triangle are truly what you have written, then we have, courtesy of Pythagoras:


No way!
Perhaps you meant to write "...and the length of legs are 3 and 4,..."
If so, then we have a right triangle.
The tangent of the three angles are:
Tangent 90 degrees is undefined.
The tangent of the other two angles, A and B, are:


2)
Find the distance between the two points (1, 3) and (5, 6)
You can use the distance formula:
Substituting , , , and




Notice that we use only the positive answer from the square root because distance is a positive (usually) value.

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Well, your error is found in the step..."completing the square"
Let's go over that:
then you said ..."complete the square", but before you do that, you should factor out a 3 then divide through by 3 and you will get:
Divide both sides by 3.
Now you complete the square when the coefficients of the and terms are equal to 1.
Factor the trinomials.
Now compare this with the standard form for a circle of radius r and center at (h, k).
and you can see that h = 1, k = -2, and
The center (h, k) is at (1, -2) and the radius is

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Assuming that there is no sag in the wire betwwen the two poles (not a realistic situation), then we can make a right triangle in which the wire represents the hypotenuse, the base of the triangle is represented by the distance between the two poles (36ft.) and the height of the triangle is simply the difference between the heights of the two poles (48ft. - 33ft. = 15ft.).
Now you can apply the pythagorean theorem to find the length of the hypotenuse (the length of the wire (W))


Take the square root of both sides.
ft.
The length of the wire is 39 feet.

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We could find the height of the resulting cone if we knew their slant-height and radius.
Starting with the circular sheet of paper whose radius (R) is 6 cm. If you divide this into three equal sectors to form the cones, then each cone will have a slant-height equal to the radius of the original circle, right?
Now the circumference of the base of each cone will be equal to one third of the circumference of the original circle.
The circumference (C) of the original circle is:
and, of course, you know that R is 6 cm, so...
Divide both sides by 3 to find one third of this.
and this is the circumference (c) of the cone base.
From this we can find the radius (r) of the cone base because but this is equal to . So we can write:
Simplifying, we get:
and so...
this is the radius of the cone base.
Now we can use the Pythagorean theorem to find the cone height using the slant-height of the cone (6 cm) as the hypotenuse of a right triangle and the radius of the cone base (2 cm) as the base of a right triangle.
where h is the height of the cone.
Subtract 4 from both sides.
Take the square root of both sides.
cm. This is the height of the cone to the nearest thousandth.

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You might ask yourself why you don't think that your answer is right?
Multiply both sides by a.
Subtract a from both sides.
Finally. add 5 to both sides.
which is precisely what you got for an answer!
This could also be written as:

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Well, you can start by assigning variables to the unknown quantities.
Let P = the number of pounds of peanuts at $2.50 per pound.
Let C = the number of pounds of cashews at $5.00 per pound.
From the problem description, you are told that P + C = 30 pounds and this mixture will cost $4.00 per pound.
We have enough information to set up the necessary equation, but there are two unknowns and only one equation. We need to reduce the number of unknowns to only one.
$2.50(P) + $5.00(C) = $4.00(30) but you are also told that:
P + C = 30 which can be written as C = 30 -P so we'll replace the C with 30-P.
$2.50(P) + $5.00(30-p) = $4.00(30) Now you can simplify this and solve for P.
2.5P+5(30-P) = 4(30)
2.5P+150-5P = 120 Combining like-terms, we get:
-2.5P = -30 Divide both sides by -2.5
P = 12 You will need 12 pounds of peanuts.
C = 30-P
C = 30-12
C = 18 You will need 18 pounds of cashews.
Check:
Cost of 12 pounds of peanuts + cost of 18 pounds of cashews = cost 0f 30 pounds of mixed nuts at $4.00 per pound.
$2.50(12) + $5.00(18) = $4.00(30)
$30.00 + $90.00 = $120
$120.00 = $120.00

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'twould be most helpful if you were to use parentheses to remove any ambiguity in your problem statement!
I was wondering too...is your problem:
? written as 1/(6f-5) = 1 or is it:
? written as (1/6f)-5 = 1
I'm going to assume that it's the second equation.
Add 5 to both sides of the equation.
Multiply both sides by 6f.

Finally, divide both sides by 36.

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Find the fifth root (not the 5th square root) of 1.63.
Most calculators have a square root key but not the cube root, fourth root, etc.
To find a higher root than square root, it is good to know that the square root of a number can be written as:

In general, the nth root of a number, x, can be written:
So you can write the 5th root of 1.63 as:
and this can be put in your calculator using the caret sign (^)
So enter the 1.63 then the caret (^) then (1/5) followed by the = or ENTER key and you should get: 1.10264959142

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If the angle is A, then its supplement is (180-A), so three times ts supplement would be 3(180-A), so you can write:
A = 3(180-A)

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Solve for h:
Subtract from both sides.
Now divide both sides by

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Well, you have given the of height (h) as a function of time (t) as:
where ft/sec, and .
So what you are asking for is at what time, t, will h = 180 ft? Right?
Substituting h = 180, you can solve for t.
Subtract 180 from both sides.
Simplify this by factoring a -4.
Factor trinomial in the parentheses.
Apply the zero products principle.
or then:
or so...
or
The answer to the question..."At what time will the arrow reach a height of 180 ft?"
It will reach a height of 180 ft in 2.5 seconds on the way up and again at 4.5 seconds on the way down.

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Here's a guess!
T, F, S, N, E. T, F, S, N, T,
What do they mean...my guess is they are the initial letters of the odd integers beginning with 3.
T = 3
F = 5
S = 7
N = 9
E = 11
T = 13
F = 15
S = 17
N = 19
T = 21

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This problem involves diophantine equations, where you have more unknowns than equations.
The solutions, however, are limited by the fact that the unknowns must be integers, i.e. whole cows, whole horses, etc.
We can set up two equations as follows: H = number of horses, C = number of cows, and ch = number of chicks.
From the problem description, we can write:
H+C+ch = 100 The total number of animals is 100.
($5)H+($3)C+($0.50)ch = $100 The total amount the farmer spends.
The two equations are:
1) H+C+ch = 100
2) 5H+3C+0.5ch = 100 Multiply this equation by 2 and subtract equation 1) from the result.
1) H+C+ch = 100
2a) 10H+6C+ch = 200
--------------------
3) 9H+5C = 100 Subtract 9H from both sides.
3a) 5C = 100-9H Now divide both sides by 5.
3b) C = 20-(9/5)H Now C can only be an integer if H is a multiple of 5. Let's try H = 5.
3c) C = 20-(9/5)(5)
C = 20 - 9
C = 11 This is one possibility. Now let's try H = 10
3d) C = 20 - (9/5)(10)
C = 20 - 18
C = 2 This is another possibility.
H cannot be greater than 10 otherwise you would end up with C being a non-integer or a negative number.
So, if C = 11 when H = 5, then ch = 100-11-5 = 84 - the number of chicks.
Let's check this solution.
H+C+ch = 100
5+11+84 = 100 and...
5($5) + 11($3) + 84($0.50) = $25 + $33 + $42 = $100 This is a valid solution!
Now if C = 2 when H = 10, then ch = 100-2-10 = 88 - the number of chicks.
Let's check this solution.
H+C+ch = 100
10+2+88 = 100 and...
10($5)+2($3)+88($0.50) = $50+$6+$44 = $100 This solution is also valid!
The two solutions are:
1) 5 horses, 11 cows, and 84 chicks.
2) 10 horses, 2 cows, and 88 chicks.

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To find the average score (A) of the first five tests, you'll add the scores and divide by 5.
where x is the score on the fifth test.
Now you need this average (A) to be at least 80 () but less than 90 ().
You can write the inequality thus:
To solve this, you'll simplify the numerator.
Now you'll multiply everything by 5.
Simplify.
Now you'll subtract 326 from everything.
Simplify.
Read this as: x is at least 74 but less than 124.
The range of scores needed to get a B average is at least 74 but less than 124.

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Let n be the required number. From the problem description, you can write:
Add the fractions, the LCD is 150.
Simplify.
Multiply both sides by 150.
Divide both sides by 43.

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1) Let x be the first odd integer, then x+2 will be the next consecutive odd integer.
From the problem description, you can write:
3x = 2(x+2)+5 Apply the distributive property to the right side.
3x = 2x+4+5 Simplify.
3x = 2x+9 Subtract 2x from both sides.
x = 9
The two integers are 9 and 11
Check:
3(9) = 2(11)+5
27 = 22+5
27 = 27
2) I don't know of an easy way to see if a similar problem to one you want help with has already been answered. You could try clicking on "Recent problems solved by all" and do a manual search but that could be quite time-consuming.

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Factor:
Do you recognise this as the sum of two cubes?
...but of course!
The sum of two cubes can be factored thus:

In your problem, A = x and B = 2, so...
and we're done!
Let's check the solution by multiplying the two factors, using FOIL.
Simplify by combining like-terms.

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From your equation, it appears that you assumed that the two segments were the same distance, but there is nothing in the problem to support this assumption.
So, back to basics:
and...

We know that:
and...

and we also know that:
So we can write:

aagh...to many variables! Let's replace with and replace with , so now we have:
and
Solve this for
...but, of course, , so...
Good, now we have one variable , so let's solve for it.


and...
hrs. Now we can find and


km and...


km
Check:
Total distance is = 90km

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Simplify:
When multiplying quantities with exponents, if the bases are the same (base is 2 in this case) then you add the exponents:
=