Systems of Equations

Katie B's Page
of
Systems of Equations

Are any of these how you feel in math class? This page will help you out some. Just read the problems, and if you have any questions, e-mail me at [email protected].

P roblem #1- Sibling Rivalry
(Word Problem)

Carlos is 8 years older than his sister Maria. Four years ago Maria was 2/3 as old as old as Carlos. How old are they now?
This is how the problem should look. How I worked it out is on the left.

C = M + 8
M - 4 = 2/3C

M - 4 = 2/3(M + 8)
M - 4 = 2/3M + 5 1/3
+4 +4
M = 2/3M + 9 1/3
-2/3M -2/3M
1/3M = 9 1/3
divide both by 1/3
M = 28

C = M + 8
C = (28) + 8
C = 36

So, Maria is 28 years old and Carlos is 36 years old .

How I Got That

First thing's first: you need to think up variables for Carlos and Maria's age. I used "C" and "M", respectively, but you can use any two letters that you want.
Now you need to make up two equations. Since Carlos' age is or equals eight more than Maria's age, I started with C=M+8.
The second equations shows the next set of information the problem gave me. "Four years ago" is -4 and "2/3 as old as Carlos" is 2/3C . Just make sure you put the right information on the right side of the equals signs and with the right variable.
Next, figure out what would be the easiest way to solve this system of equations. Since the "C" is all by its self, let's attack it. So substitute the "C" into the second equation and do the math. Now all you have is a simple algebra equation. Add the negative 4 to both sides, then subtract the -2/3M from both sides. Divide both sides by 1/3 and you have what "M" equals. Then plug "M" back into the first equations and you find "C". sghg

Very good! Now let's try another type of solving.

Okay, so now you're starting to catch on? Try this next one that involves substitution.
Problem #2- Substitution The Equations given: Y = 3X X + Y = 8 X + (3X) = 8 4X = 8 divide by 4 X = 2 (2) + Y = 8 Y =6 So the answer is (2,6)	How I Got That Okay, substitution is so easy it's almost fun. Since they tell you that Y= 3X, all you have do is substitute that into the 2nd equation and then it's smooth sailing. Combine like terms, then divide by 4. All you have to do now is put the answer you just got into one of the equations (it doesn't matter which one) and find what Y equals. *TahDah**! Have fun!

This is the last method! You're doing great, so don't stop now!
P roblem #3- Linear Combination The Equations: 2X + Y = 6 -2X + Y = 20 Just add them together! 2Y = 26 Y = 13 Now use substitution. 2X + (13) = 6 -13 -13 2X = -7 X = -3.5 So the answer is (-3.5, 13)	How I Got That Linear Combination is also pretty simple. These kinds of problems are easiest when you have two variables that cancel eachother out, like 2X and -2X. When you add them together, it leaves you with only one variable, so all you do is divide. Then apply the above method and substi- tute the first answer into the next equation. And that's all, folks!

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