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To solve equations of this kind you have to think of the variable expession 4X (or whatever) as a number. With an equation you can subtract or add a number to both sides of the equation and it still remains true that the two sides are equal. If you saw 2 + 7 = 9 you know you could subtract 2 from each side and end up with 7 = 7. Well, instead of 2 you have something like 4X. That would be 4X + 7Y = 32. Let's subtract the number 7Y from both sides. You get
4X + 7Y - 7Y = 32 - 7Y. This simplifies to 4X + 2Y = 32 - 7Y. Subtract the number 2Y from both sides, giving 4X = 32 - 5Y. Aha! you now know what 4X equals. Since it's the same X and Y in the second equation, which is 4X + 2Y = 64, you can subtract 4X from both sides of the second equation. But here's the trick ... you use 4X when you subtract from the left side of the equation and you use 32 - 5Y when you subtract from the right side of the equation. They are equal, so you can do this. You are subtracting the same number from each side of the second equation.
Doing this you get 4X - 4X = 64 - 32 + 5Y. And that simplifies to
0 = 32 + 5Y. Or, rearranging, 5Y = -32.
If 5 times Y equals -32, then 1 times Y must equal -32/5. [Do you see why?]
Now you can replace Y with its value.
4X + 2Y = 64 becomes . This is 4X -64/5 = 64.
Dividing both sides by 4 gives X - 16/5 = 16. And X = 16 + 16/5, which is
X = 96/5.
Checking . Right!
And . Right!
Of course there is a short cut, once you understand where this came from.
You can subtract the entire left side of the second equation from the left side of the first equation, and the entire right side of the second equation from the right side of the first equation. You get
4X + 7Y - (4X + 2Y) = 32 - 64
4X + 7Y - 4X - 2Y = 32 - 64
5Y = -32
Y = -32/5
This is the same thing, but just hides all the rules being applied to get there. Practice will make this easy.

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What you need to know is that the time it takes to travel over a distance is given by .
This problem has three distances that are given as a shorter distance, a longer distance that is the short distance plus 96, and the total distance which (of course) is their sum.
Let x be the shorter distance. The time it took to travel over x was . The time it took to travel over x + 96 was . The time it took to travel over the entire distance was .
Of course, the time for the total distance must equal the sum of the times for each segment, so

You solve this for x.
[get common denominator]
[simplify]
[multiply both sides by both denominators]
[simplify]
[divide both sides by 20]
[divide both sides by 3]
So, x+96= 128, and the total distance is 160.
Check by calculating the time taken for each segment and for the total:
32/60 = .5333333333...
128/120 = 1.06666666...
160/100 = 1.6
.5333333... + 1.06666666... = 1.6 [think of it as .5 and 1/3 of a tenth plus 1.0 and 2/3 of a tenth, which is 1.5 plus a whole tenth]

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See the answer to 21648.

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The slope-intercept form for the equation of a line contains two pieces of information: the slope and the y-intercept.
The slope tells how y changes when there is a change in x. For example, a slope of 3 means that for each change in x of 1 unit, y changes 3 units. A slope of -3 would mean that for each change in x of 1 unit, y changes -3 units. This is written algebraicly as y = mx, where m is the slope. y = 3x is the equation of a line that has a slope of 3. Look at a little table for various values of x:
y.....x
3.....1
6.....2
9.....3
Each time x changes by 1 (1 to 2, 2 to 3), y changed by 3 (3 to 6, 6 to 9). This is because y is always 3 times whatever x is.
Next, the y-intercept tells at what point the line crosses the y axis. That is, what is the value of y when x is zero? Well, since this is a single fixed place on the y-axis the equation of y = a gives that information. If the line crosses the y-axis at 2, then the equation would be y = 2. If it crosses at -7, then the equation would be y = -7.
The combination of these two pieces of information gives an equation of .
For the information you had (slope of 5 and intercept of -8) the equation is
y = 5x - 8.
Simple, no?

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Actually, they said no such thing! But if they had.....
Let's use two letters for the unknown weights of Tweedledum and Tweedledee. To keep them straight, use the LAST letter of their names: m and e.
So,
e + 2m = 361
This is the first statement by Tweedledum: "your weight" [e] plust twice MY weight [2m] is 361.
The second statement, likewise, is rendered:
m + 2e = 362
Now just solve.
e + 2m = 361
2e + m = 362
2e + 4m = 722 [multiplying the first equation by 2]
3m = 360 [subtracting the second equation from the third one]
m = 120 [dividing both sides by 3]
e + 240 = 361 [substituting the value of m in the first equation]
e = 121 [solving for e]
Checking:
121 + 240 = 361 [first equation]
242 + 120 = 362 [second equation]
They don't seem very fat!

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I assume your equation pairs read across. Let's take the first one. Elimination means eliminating one of the unknown variables by getting a common coefficient for it in both equations and then subtracting the two sets of equals from each other. [Remember: equals subtracted from equals gives equals.]
We have
3x + 2y = -24
4x + 3y = 18
We can make the y terms have the same coefficient by multiplying the first equation by 3 and multiplying the second equation by 2. That will make both equations have 6y as the y-term.
This gives
9x + 6y = -72
8x + 6y = 36
Now we subtract the second equation from the first.
This leaves
x = -108
Since we now know x, we can put that value into either of the original equations and solve for y.
3(-108) + 2y = -24
-324 + 2y = -24
2y = -24 +324 = 300
y = 150
Checking with x=-108 and y=150 shows this to be correct.
Now you do the second set. [Hint: multiply the first equation by 6 and the second one by 4 and both equations will have 24 for the coefficient of y. In this case one will be positive and the other negative so you can ADD the equations together to get rid of the y-term.]

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Like all algebraic expressions you can only "solve" it when you know a value for the unknown, in this case C.
You do not have an equation; you have an expression.
You CAN, however, simplify an expression.
In this case it doesn't help much since the simplification results in
and that's it!

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This is an expression, meaning it is a sequence of operations and operands. It is algebraic since it contains a number that is unknown and is represented by x.
All the other operands (numbers in the expression) can be used in the operations to find out what they are equal to. For example, 3 + 5 has two operands (3 and 5) and one operation ( + ) and the expression 3 + 5 is equal to 8.
You have
-8x + -(5+3x) - 2
-8x means the negative of 8 times x (whatever number x is, which we don't know!)
-(5+3x) means the neagtive of the quantity 5+3x, and you know what 3x means, right?
and 2
the first number [-8x] is added to the second number [-(5+3x) and then the third number [2] is subtracted from that sum.
This can be simplified by applying the negative of the quantity (5+3x) giving
-5 -+3x. It is usual to write a negative addition as a subtraction, so this becomes -5 - 3x. The whole expression is now
-8x +-5 - 3x - 2
Again, the addition of a negative is usually written as a subtraction, so:
-8x - 5 - 3x - 2
Now terms can be grouped and rearranged.
-8x - 3x - 5 - 2
And arithmetic can be done, giving
-11x -7
That's the simplified expression which will have the same value as the original.
Of course, the value of the expression is unknown until you know what value x represents. Example: if x=3, the expression becomes
-11(3) -7 == -33 - 7 = -40.
Hope that helps.

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First, what are two consecutive even integers. If you let x be the first one, what would the next even integer be? If you said x+2 you would be right.
So, the smaller of the two is x, the larger is x+2.
Now you know that 3 times the larger one is equal to the smaller one less 10.
3 times the larger one is written algebraicly like this: 3(x+2).
The smaller one less 10 is written like this: x - 10.
You know 3(x+2) = x-10.
Can you solve this equation for x?
3x+6 = x-10 [distributive rule]
2x+6=-10 [subtract x from both sides]
2x=-16 [subtract 6 from both sides]
x=-8 [divide both sides by 2]
x+2=-6
So three times the greater number (-6) is equal to the smaller number (-8) minus 10.
[Checking: 3(-6)=-18. -8-10=-18.]

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This is an example of using the distributive property: a(b+c) = ab + ac.
8p -5(p + 3) = (7p - 1)3
8p - 5p -15 = 21p -3
Now it's a matter of rearranging terms (addition and subtraction are commutative).
3p - 15 = 21p - 3
- 15 + 3 = 21p - 3p [adding 3 to both sides and subtracting 3p from both sides]
-12 = 18p
p = - 12/18 = - 2/3
HTH

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Everything you posted looks right. Remember:
a(b+c) = ab + ac.
That's it.

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The underlying meaning of a determinant is complex and cannot be covered here. It is a useful value to know when working with matrices. There are a couple of ways of calculating the determinant of a matrix, and for a 3 x 3 there is a relatively easy procedure. I'm surpised you haven't been given it (if you haven't) but I'll try it here.
You first get the products of all the left-to-right diagonals of the matrix and add them up. You then get the products of all the right-to-left diagonals of the matrix and add them up. You then subtract the right-to-left sum from the left-to-right sum. The diagonals are all referenced from top to bottom.
You can see how this is a very poor way to describe the procedure. Let's make it mathematical.
We will reference the elements of the matrix with the following notation:
will mean the element in row x and column y.
In a 3 x 3 matrix the upper left element will be and the lower right element will be . Get it?
Okay. The left-to-right diagonals are as follows:
, , ; , , ; and , ,
Whew! For the matix values you gave, this amounts to
(3 x 5 x 1), (-2 x 3 x 2), and (0 x -1 x 7). Do the multiplications and add them up.
15+(-12)+0 = 3. This is the value of the products of all the left-to-right diagonals.
The right-to-left diagonals are as follows:
, , ; , , ; and , ,
For the matrix values you gave, this amounts to
(0 x 5 x 2), (-2 x -1 x 1), and (3 x 3 x 7). Do the multiplications and add them up.
0+2+63 = 65. This is the value of the products of all the right-to-left diagonals.
Subtract the right-to-left from the left-to-right.
Determinant = 3 - 65 = -62
CHECK MY WORK!!!

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Ah, if I could get rid of my cares so easily! [watch those typos]
It really doesn't matter if they are going North, South, or any other direction. The question boils down to: two cars travel. One goes at a speed of ? miles per hour. Let's use the letter x for that unknown speed. The other one goes x+10 miles per hour. [Do you see why?] After three hours of travel, the total distance they have traveled is 354 miles.
The slower car goes 3 hours times x miles per hour, which is 3x miles.
The faster car goes 3 hours times x+10 miles per hour, which is 3(x+10) miles.
The total distance they travel is 3x + 3(x+10) = 354 miles.
This is easy to solve.
3x + 3x + 30 = 354 [simplifying the expression]
6x + 30 = 354 [more simplifying]
6x = 324 [subtracting 30 from both sides]
x = 54 [dividing both sides by 6]
x+10 = 64
Checking: 3(54) + 3(64) = 162 + 192 = 354

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I would like to add that the phrase "greater or less than" is referring to those values of k which make the inequality true. By solving for equality you have a value for which "greater or less than" is NOT true. So, the correct answer is any value EXCEPT k=2.

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Read the answer to 14607 and think it over.

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Read the answer to 14607 and think it over.

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Read the answer to 14607 and think it over.

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Read the answer to 14607 and think it over.

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Read the answer to 14607 and think it over.

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What do you want to know? I assume you wonder what values of t make the statement true!
Try the following:
If t=4, then 14t=56. What happens if t gets smaller, say 3.999. 14t will then be smaller than 56.
What happens if t gets larger, say 4.00001. 14t will then be larger than 56.
Any number larger than 4 would then make the inequality 56<14t true.
So an "answer" would be: 56<14t for all values of t>4.
I suspect that is what you were looking for.
NOTE: while this is all intuitive, you can be more formal.
56 < 14t
4 < t [dividing both sides by 14]
This says the same thing. The only thing to remember about inequalities is that you can add, subtract, multiply and divide by equals on both side BUT MUST CHANGE THE DIRECTION OF THE INEQUALITY if you divide or multiply by a negative number.
3 > 2 [true]
6 > 4 [multiplying both sides by 2]
BUT
-6 < -4 [multiplying both sides by -2]
Play with this a bit with simple numbers (and draw them on a number line) until you see how this works.

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Okay, picture the rectangle. The length of one side is x, i.e. a value we don't yet know so we use a letter to indicate that value. The other side is twice that length. So, the other side is 2x. When you add up the sides (x + 2x + x + 2x) you get a value of 480. Putting this into an algebraic expression (instead of words) gives . This can be solved easily.
6x = 480
x = 80 [dividing both sides by 6]
That is the length of one side.
2x = 160 [multiplying both sides by 2]
That is the length of the other side.

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What you have is
What this means is: the value of y is equal to three times the value of x cubed, which is x times x times x, less the value 10.
If x = 1, for example, you multiply 1 three times (1 x 1 x 1 = 1), then multiply that value by 3 (3 x 1 = 3), then subtract 10 from that value (3 - 10 = -7). So, y=-7 when x=1.
The order of operations, when there are no parentheses used to indicate a change in this order, is do all powers and roots, then all multiplication and division (left to right), and then all addition and subtraction. This case has power first (cubing x), multiplication (3 times that), and subtraction (- 10).
Try it for x=2: 2 cubed is 8 [do you see why?], 8 times 3 is 24 [last time I looked], and 24 - 10 is 14. So y=14 when x=2.
Now do the rest!
NOTE: If the problem presenter had wanted you to multiply the value of x by 3 FIRST, and then do the cubing it would have been written as follows:

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The idea is this: the points that are a distance of 2 from the origin forms a circle with radius 2 around the origin. The equation for that circle around the origin is because the squares of the two sides formed by the x and y coordinates add up to the square of the hypotenuse.

In the given case, since the x and y are known to be equal, then you have
which means and . So,
Since the square root can be plus or minus, the only question remaining is which is it? The answer lies in the second quadrant, which means the x value is negative and the y value is positive. So the coordinates of a point with the absolute values of the x and y values being equal that lies 2 from the origin are ,

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No, you have a simple error. If y is the number of tickets sold for couples, then 2y persons attend for those tickets. So, x + 2y = 128. Your equation for the dollar amount of x and y ticket sales is correct.
Solve x + 2y = 128 and 20x + 35y = 2280.
[I got x=16 and y=56]

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Your first job is to calculate the slope of the line. Because you know two points, you can find the rise between them (difference in y's) and the run between them (difference in x's). A simple division of rise/run gives you the slope.

So m = -9/7.6
Now you have the equation
Now find b. We know that when x=0 y=b. So there is a point (the y-intercept) on the y-axis at b. The slope between the points (-3.5, 4.7) and (0, b) is -9/7.6 (the slope is the same everywhere).
So,
Solve this for b.


[do the math!]
This gives the equation

Check this by putting in each value of x and see that you get the correct value for y.

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This is just a bit of practice in converting the words into algebraic statements. Take them slowly.
Let's call the angles FA, SA, and TA (first angle, second angle, third angle).
The FA is equal to some degrees. Call it x.
So, FA=x. Simple.
The second angle is 10 less than twice the first. Well, twice the first is 2FA and 10 less than THAT would be 2FA - 10.
So, SA = 2FA - 10. Simple again.
The third angle is 25 more than the second.
So, TA = SA + 25. Still simple.
The hard part (?) is getting everything in terms of x.
Look at SA = 2FA - 10. See FA? Well, we know it is equal to x (because we said so). Then SA = 2x - 10. See why?
Now look at TA. It's SA + 25. But SA is 2x - 10.
So, TA = 2x - 10 + 25. What we now have is
FA = x
SA = 2x - 10
TA = 2x + 15 [I did the math]
And lastly, we know that x + 2x - 10 + 2x + 15 = 180! [Sum of the angles]
Can you simplify that equation to find x?
[You should get an answer that x = 35.]
Then calculate the other angles using the value you found for x.

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Assume you know where they will meet and call it M. How long does it take Mary to get to M? The answer is 21 mi/hr times x hours. So, we have
21x = M
How long does it take Sally to get to M? The answer is that she must travel the distance that takes her to M starting at 60 mi. And that is 60-M. So, we have
15x = 60-M
Substituting the value of M from the first equation into the second we get
15x = 60 - 21x
36x = 60
x = 5/3
This says that after 5/3 of an hour Mary will have travelled 35 miles (21 x 5/3) and
Sally will have travelled 25 miles (15 x 5/3).
Checking: 35 miles + 25 miles = 60 miles.

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You are finished.

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Consider what the function is if you let x=0. It graphs to the points on the line at y=1. (I'll assume you accept that y=f(x)=x+1.) But you can only allow x to take on values of less than 0 (the negative numbers), so the points of y will be all the points below the line for y=1. The range is the area below the line y=1, but not on it. And certainly not above it! To get there, x would have to be a positive number.
Now, you are only allowing negative integers in the domain. That means that the value of y will also be an integer. So, only the points below the line y=1 that have values of y that are integers are in the range. That would be the lines y=0, y=-1, y=-2, etc.
Done.

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I am going to assume that you know the area of a circle is
So, if you know the r (radius) you can calculate the area. Sometimes it is better not to multiply out by (3.14159) but just leave it as .
Your first question has a diameter of 4 on the outside, and a diameter of 2 on the inside. If you could find the areas of both of these circles, could you find the area of the ring? Try subtracting the area of the inner circle from the area of the outer circle!
Also, remember the diameter is twice the radius. So, putting this all to work:
The area of the outer circle is
The area of the inner circle is
The area of their difference is
Your second question is similar except here you are dealing with the circumference of a circle, which is
You know what r is (6), so how far will the tip move in one hour? This is the circumference of the circle with a radius of 6. [Do the math]
But we want to know how far it moves in 36 minutes. Well, the hour is 60 minutes, so the part of the hour that the tip moves is 36/60 of the whole circumference. [Do the math] Do you get ?

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Assume you know where they meet. It is at point M. They take the same amount of time (x) to travel to point M.
She travels at 2 mi/hr. In x hours she will travel 2x mi. This is the distance M. So, 2x = M.
He travels at 3 mi/hr. In x hours he will travel 3x mi. This is the distance from where he is to M. But he is already at 4 mi. (That's how far apart they are when they start!) So the distance he travels to get to M is 4-M. [Can you see why?]
So, 3x = 4-M.
Now we solve for x.
2x = M
3x = 4 - M
3x = 4 - 2x [substituting 2x for M]
5x = 4
x = 4/5
Check: In 4/5 of an hour she will travel 8/5 mi.
In 4/5 of an hour he will travel 12/5 mi.
The total distance they traveled is 8/5 mi. + 12/5 mi. = 20/5 mi. = 4 mi.