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Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
What is the process of finding the solution of an equation?
A system of equations simply means, what number can you plug in for X and Y that will make both equations work?
For example,
Solve the following system of equations:
x + y = 11
3x - y = 5
In order to solve a system of equations, you have 2 ways.
Method 1: Substitution
Choose 1 equation. It doesn't matter which!
To make it easier, pick an equation without alot of numbers multiplied to X or Y.
ie: x + y = 11.
Now, solve for X or Y. You pick! OK, I'll pick. (but it doesn't matter which. just pick either!)
I'll solve for X.
x + y = 11
Subtract y from both sides, because I want my equation to look like x = something.
x = 11 - y
Now that we've solved for a variable (x), plug what x is equal to (11-y) into the equation you didn't solve, wherever you see x. Make sure to use parenthesis!
3x - 5 = y
3(11-y) - 5 = y
Now use your algebra to simplify the equation.
33 - 3y -5 = y (distributive property -- this is why we use parenthesis....)
28 - 3y = y
28 = 4y
7 = y
Now that we have a number for Y, plug 7 in wherever you see Y into one of your original equations. We're trying to solve for X now. In fact, to make it easier, you can plug y = 7 into the equation we solved x for!
Remember, x = 11-y ?
x = 11 - (7)
x = 4
So we know y = 7 and x = 4. These numbers should work in BOTH original equations!! If they don't, you made a mistake and you should double check your algebra.
x + y = 11
(4) + (7) = 11
11 = 11 (check)
3x - y = 5
3(4) - (7) = 5
12 - 7 = 5
5 = 5 (check)
Method 2: Elimination
As this method suggests, we need to "eliminate" or get rid of 1 variable.
Line both equations up so the x's and y's are stacked on top of each other, and the = signs as well. Like you were going to do an addition, or subtraction problem.
x + y = 11
3x - y = 5
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What our goal is, is to eliminate one variable. Just like in method 1, you choose which variable. The easiest way is to pick a variable with no numbers in front.
Those y's look pretty good!
When using the elimination method we want 2 things to happen.
1) The number in front of the variable being eliminated is the same in both equations
2) The variable being eliminated must have opposite signs in both equations.
Luckily, our y variable has no coefficient (number in front) AND already has an opposite sign! (+y on top, -y on bottom). All we need to do now is add downward, like a regular addition problem.
1x + 1y = 11
3x - 1y = 5
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4x = 16
Notice the y's cancelled each other out? They "eliminated" themselves, which was our goal! Remember to add the numbers on the other side of the equals sign as well.
Now just solve for X.
4x = 16 -> x = 4
Once we find a value for x, substitute x=4 into either original equations, to solve for y.
x + y = 11
(4) + y = 11
y = 11 - 4
y = 7
So x =4 and y=7, just like in the first method. Use whichever makes you feel comfortable. I find substitution is much easier, but you might prefer elimination. Good luck!
How do you work out the perimeter of a rectangle?
Take the length of the rectangle and multiply it by the width.
For example, if I had a rectangle with two sides 4 inches in length and two sides 3 inches in length, then the area would be 4x3, which equals 12 square inches.
What is the maximum number of distinct edges in an undirected graph with N vertices?
Let G be a complete graph with n vertices. Consider the case where n=2. With only 2 vertices it is clear that there will only be one edge. Now add one more vertex to get n = 3. We must now add edges between the two old vertices and the new one for a total of 3 vertices. We see that adding a vertex to a graph with n vertices gives us n more edges. We get the following sequence
Edges on a graph with n vertices: 0+1+2+3+4+5+...+n-1. Adding this to itself and dividing by two yields the following formula for the number of edges on a complete graph with n vertices: n(n-1)/2.
Why is the square root of three an irrational number?
I'm assuming that you mean 'square root'. Yes, this sum is irrational. So are each of the two numbers alone. A simple proof can be done by writing x=square root 2 + square root 3 and then "squareing away" the square roots and then use the rational roots theorem. The sum or difference of two irrational number need not be irrational! Look at sqrt(2)- sqrt(2)=0 which is rational.
WHOEVER ANSWERED THIS PREVIOUSLY HAS NO IDEA WHAT WE ARE ALL TALKIN ABOUT , OBVIOUSLY
* * * * *
Ok, here is someone who might know.
If the first number is bigger than the second, the difference is positive;
If the first number is the same as the second, the difference is zero; and
If the first number is smaller than the second, the difference is negative.
How many digits of pi have been found?
Chuck Norris counted to infinity -twice!
Only Chuck Norris can divide by ZERO!
The term "digit" is from the Latin digitus meaning "finger or toe". Therefore there is no such thing as a three digit number, even in pi. And I like pi better than cake. Mmhh...pi!
Is the converse of a true conditional statement always false?
No. Consider the statement "If I'm alive, then I'm not dead." That statement is true. The converse is "If I'm not dead, then I'm alive.", which is also true.
Can the product of two mixed numbers be a whole number?
It's always a common multiple; it's not always least.
Simple counter example:
4 × 6 = 24
But LCM(4, 6) = 12
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Note: HCF(4, 6) = 2
What is true of any two whole numbers that the product of the two numbers is equal to the product of their highest common factor and lowest common multiple.
eg 4 × 6 = hcf(4, 6) × lcm(4, 6) = 2 × 12 = 24.
Easy. And it need not even involve esoteric mathematical formulas.
Find a circle. Measure the diameter. Write that down, and let us assume for example that it's ten meters.
Now measure the circumference. I know, the meter stick is straight, but that's okay. Get a string and put it all around the circle. Make a mark on the string at the point where it comes all the way back around.
Now straighten the string and measure that. You'll note that it is 31.4 meters! And if you measured even more accurately, you'll note that it's 31.41 meters! Which when you divide by the ten meters, equals 3.14!
But what if it's not ten meters in diameter, where's the proof then?
Well, lets say you find a circle five meters in diameter. Measure the circumference and you'll find it's 15.7 meters. Now take 15.7 and divide it by 5. It will equal 3.14!
No matter what circle's circumference you measure, if you divide that by the diameter, it will equal 3.14. If your measurements are more accurate, it will always equal 3.141. More accurate? 3.1415. And so on.
What is the angle bisector concurrency theorem?
the definition of an angle bisector is a line that divides an angle into two equal halves. So you need only invoke the definition to prove something is an angle bisector if you already know that the two angles are congruent.
If you are born in 1988 how old are you now?
If you are born in 1988 you would be 24 in 2012. To find anyone's age, take the current year and subtract the year that the person was born in.
What is the value of one ounce of 999 fine silver today?
The value of silver and other metals constantly changes so any answer posted here would be out of date almost immediately. While it's not normal WikiAnswers policy to say "use the Internet", that's the best approach in this case. You can check a site such as www.kitco.com, CNNMoney, etc. for the latest prices.
Such statements are called postulates in geometry and axioms in other areas.
Definitions are also accepted without proof, but technically they are abbreviations rather than statements.
How many multiples of 3 are there between 1 and 1001 and show work?
I would do it like this though I'm sure there are other ways:
There are 10 multiples of 3 every 30 (3,6,9,12,15,18,21,24,27,30).
There are (1000/30)= 33 lots of 30 in 1000
Therefore there are (10 x 33)=330 multiples of 3 which brings us up to 990 (330x3 =990)
Then there are 3 more multiples of 3 993, 996,999
This makes the total 333 multiples of three between 1 and 1001
What is the sum of the prime numbers between 40 and 60?
The prime numbers between 40 and 60 are 41, 43, 47, 53 and 59. These sum to 243.
State and prove Lusins theorem?
Lusin's theorem says that every measurable function f is a continuous function on nearly all its domain.
It is given that f measurable. This tells us that it is bounded on the complement of some open set of arbitrarily small measure. Now we redefine ƒ to be 0 on this open set. If needed we can assume that ƒ is bounded and therefore integrable.
Now continuous functions are dense in L1([a, b]) so there exists a sequence of continuous functions an tending to ƒ in the L1 norm. If we need to, we can consider a subsequence.
We also assume that an tends to ƒ almost everywhere. Now Egorov's theorem tells us that that an tends to ƒ uniformly except for some open set of arbitrarily small measure. Since uniform limits of continuous functions are continuous, the theorem is proved.
What is the LCM and GCM 46 50 and 4?
LCM(46, 50, 4) = 2300.
There is really no such thing as a "greatest common multiple" (GCM). Once you find the least common multiple of a set of numbers, you can keep adding the LCM to itself over and over again. Each new number you get will be a common multiple of your set of numbers, but each new number will always be larger than the previous. This means that you can keep adding while the number approaches infinity and you will still never find a greatest multiple.
What is a negation of a statement?
Negation says you should write the opposite.
So let's take the statement, Today is Monday. The negation is today is NOT monday. Sometimes it is harder.
Say we have the statement, EVERY WikiAnswers.com question is a great questions. The negation is Some questions on WikiAnswers.com are not great questions.
How do you prove that the diagonals of an isosceles trapezoid are congruent?
Suppose the diagonals meet at a point X.
AB is parallel to DC and BD intersects them
Therefore, angle ABD ( = ABX) = BAC (= BAX)
Therefore, in triangle ABX, the angles at the ends of AB are equal => the triangle is isosceles and so AX = BX.
AB is parallel to DC and AC intersects them
Therefore, angle ACD ( = XCD) = BDC (= XDC)
Therefore, in triangle CDX, the angles at the ends of CD are equal => the triangle is isosceles and so CX = DX.
Therefore AX + CX = BX + DX or, AC = BD.
Proof that grahams number is the largest number?
No such proof can exist within consistent axioms - to add 1 to Graham's number would generate a larger number, and so, by this counterexample, Graham's number cannot be the largest.
It is the largest used in a mathematical proof, because all proofs have been noted in a proof 'by exhaustion', in which all cases (in this case, proofs) have been verified to have smaller constants .
Archimedes made many things. He made (invented) the Archimedes screw which helps farmers with their irrigation. He also supposedly made Archimedes Death Ray (which is a myth) and Archimedes Claw which is said unrealistic by modern engineers.
