(and Jane Rowden too!)
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Welcome to the culmination of world
civilization (math + The Beatles = a web site that can't get any
better!)! Here, not only will you find pictures and lyrics of
everyone's favorite band, but you will also (hopefully) learn how to do
linear combination, substitution, graphing, and a word problem
involving linear equations. |
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| HELP! When I was younger, so much
younger than today, I never needed anybody's help in any way. But
now those days are gone I'm not so self-assured. Now I find, I've
changed... |
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| The
first method we will conquer is LINEAR COMBINATION! Here's our problem: Find x and y if: -2x+3y=20 4x+4y=-15 Now, The Beatles and I know what you're thinking. You're thinking: "HELP! I NEED SOMEBODY! HELP! TWO VARIABLES! WHATEVER AM I GOING TO DO!" Let it be, man! Chill out! We're here to help! We'll go step by step: 1) The objective of this method is to combine the two equations to completely eliminate one variable so all we have left over is a solvable one-variable equation. We do this by multiplying one or both equations by a number that will eliminate the said variable. In this problem, we are going to multiply the first equation by 2 to eliminate x. 2(-2x+3y=20) and the result is -4x+6y=40 2)Now we take this new and improved equation, and merge it with the other equation like so: -4x+6y=40 4x+4y=-15 10y=25 3)YAY! Now we have a simple one variable equation! Divide both sides by 10 to get y. y=25/10 or 5/2 4)But we are not quite finished. The problem asks for y AND x. To get x, we "plug" the 5/2 into either of the original equations (we will use the second one) in place of y like so: 4x+4(5/2)=-15 and the result is another one variable equation (don't forget to simplify) 4x+(20/2)=-15 4x+10=-15 5)Now subtract 10 from both sides 4x+10=-15
-10 -10
4x=-25
6)Now divide both sides by four 4x/4=-25/4 and our answer is: x=-25/4 and our final answer is:
(-25/4, 2.5)
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| In the town
where I was born lived a man who sailed to sea, and he told us of his
life in the land of submarines... We all live in a yellow submarine...
A yellow submarine... |
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| The
next method we are going to glide through is SUBSTITUTION! Here's our problem: x - y = 17 1/2x - 3y = 1 1)The objective of this method is to determine a value of one variable in terms of the other variable and plug that value into one of the equations. We are going to solve for y in the first equation by subtracting x from both sides and dividing both sides by -1. x
- y = 17
-x
-x
-y/-1 =
(17-x)/-1
y = x - 17 2)Now that we have our y value, we are going to substitute it in for y in the other equation, simplify, and solve for x like so: 1/2x - 3 (x - 17) = 1 1/2x - 3x + 51= 1 combine like terms -5/2x + 51= 1
-51
-51
5/2x
= -50
now divide multiply both sides by 2 and divide by -5 (-5/2x)2
= (-50)2
(-5x)/-5 = (-100)/-5 x=20 3)YAY! We have our x value! Now we take 20 and substitute it in for x in one of the above equations (we will do the first equation) and solve: (20)
- y =17
-20
-20
-y = -3
y = 3 and our final answer is (20,3)  |
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| Blackbird
singing in the dead of night, take these broken wings and learn to
fly. All your life, you were only waiting for this moment to
arise... Blackbird, fly... Blackbird fly... |
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The third and
final method we will discover is GRAPHING!
Here's the problem: Find x and y if: y=2x+5 y=-2x-3 1)This method is really simple, so the first step is to breathe in and peace out. 2)Now that you are sufficiently centered, take a look at the first equation. Since it is a line, it should graph like a line too, right? Try it.  What do you know? It does! 3)Graph the other equation.  4)When you graph this on a sheet of paper, you should graph the two lines on the same plane (this is something I could not accomplish on the computer). The point where the two lines intersect is the answer. In this problem it is (-2, 1)  |
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| She was a Day Tripper! A
Sunday Driver, yeah. It took me so long to find out, but I found
out...........................................(guitar
solo)...................................................... |
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| After all this
helter-skelter explanation, here comes the crowning moment- The
Word Problem. Here goes nothing:  The Fab Four played their first
two concerts on opposite sides of  the world: one in Beijing, China and one in Liverpool, England. They charged the equivalent of $5 for concert-goers in Beijing, but because Liverpool is their hometown, they only charged $3. Their total gross from their first two concerts is $3943 and a total 937 tickets were sold (remember, they weren't terribly popular yet). How many people attended each concert? 1) Don't freak out because of the words that are accompanying the numbers. 2)Write the equations: if the tickets in Beijing were $5 (let's call this 5B if B stands for # of tickets sold in Beijing), and the tickets in Liverpool were $3 (let's call this 3L if L stands for # of tickets sold in Liverpool), and the total was $3943, then the equation must look like this: 5B+3L=3943 The second equation is for ticket totals and looks like this: B+L=937 3)Now we have two equations and we can pick any method of the three (I am going to do elimination) (B+L=937)X-3 -3B-3L=2811 5B+3L=3943 (2B=1132)/2 B=566 4)With a little logic, we can tell that if they only sold 937 tickets and 566 of them were sold in Beijing, then only 371 of them were sold in Liverpool, and that is our answer. |
