Hello Algebra

HELLO ALGEBRA!!!!

Hi!!! Welcome to my page!! I did this page for a project for my Algebra two class. My goal is for you to understand how to solve equations three ways, substitution, linear combination, and even word problems.
Well I hope you enjoy your time here!!!

Substitution Property

Let this cute kitty lead the way to a better
understanding of substitution!

The substitution method is a quick and easy way to solve systems of equations and here's how it works!!!

Substitution Example:

Here's the problem:
2x + y = -6
3x + y = -10

The first thing we should do is choose a variable that we can solve for or in other words "kill" a variable. Let's solve for "y" (kill x). For this step we will only use our first equation. It will now look like this.
y=-2x-6

     Now we can use the second equation (3x+y=-10) and substitute "y" (-2x-6) from the first equation into it. Then we can solve for "x".
                  3x+(-2x-6)=-10
                          x-6=-10
                            x=-4

     Now that we have "x" we can substitute it in to one of our equations so that we can solve for "y".
                    2(-4) + y=-6
                      -8 + y=-6
                          y=2
     Now let's plug in our answer and see if it's right!!
                   2(-4) +2=-6
                        -8 +2=-6
                          -6=-6
        It's right! The point is (-4,2)

Now it's time to try it out on your own!!	Check your answers!!!
2x + y = 6 3x + 4y = 4	(4,-2)
y = 5 - 4x 3x - 2y = 12	(2,-3)
3x - 4y = -15 5x + y = -2	(-1,3)
x = 3 - 3y 4y = x + 11	(-3,2)
y = -x + 6 x - 2y = -6	(2,4)

Linear Combination

Kitty will show you linear combination made easy!

Linear combination is a neat way two take two equations and find their variables.

Linear Combination Example:

Here's the Problem:
x + 2y = 6
5x + 3y = 2

           The first thing we should do is "kill" a variable. The easiest one is "x" so let's solve for "y". We do this by multiplying the first equation by -5 so that when we add the two equations together the "x's" will be canceled out.
                  -5(x+2y)=(6)
                    -5x-10y=-30
         Now let's add are two equations together and solve for "y".
                   -5x-10y=-30
                         5x+3y=2
                         -7y=-28>
                          -7   -7<
                            y=4
         Now that we have found out what "y" is let's plug it in to one of our equations and solve for "x".
                  x + 2(4)=6
                       x+8=6
                         x=-2
         Now, let's plug in our answers and check our work!
                   -2+2(4)=6
                       -2+8=6
                         6=6
             It's right!!! The point is (-2,4)

Now it's time to try it out on your own!!	Check your answers!!!
3x + 3y = 15 2x + 6y = 22	(2,3)
3x - 5y = 13 x - 2y = 5	(1,-2)
7x + 2y = -1 3x - 4y = 19	(1,-4)
2x + 3y = 7 3x + 4y = 10	(2,1)
5x - 3y = 16 4x + 5y = -2	(2,-2)

Word Problems

Solving word problems isn't hard so have no fear your heart will go on!!!!

The hardest thing about word problems is finding out exactly what they mean and putting them into a mathematical language. Once you have done that, it's a snap.

Word Problem Example:

Here's the problem: One day a store sold 45 pens, one kind at $8.50 and another kind at $9.75. In all, $398.75 was taken in. How many of each kind were
sold?

           The first thing we need to do is to get are two equations. We are going to let "x" be equal to the amount of the first set of pens and "y" be equal to the amount of the second. So let's turn are information into a mathematical language!!! We know that the total amount of pens sold equals 45 so in math lingo we can say x+ y=45 . THe first type of pens (x) was sold at $8.50 and the second type (y) was sold at $9.75 and the total amount maid from the two was $398.75 this means 8.50x +9.75y=398.75 .
Now we have are two equations and all we have to do is solve for "x" and "y"!
                              x+ y=45
              8.50x +9.75y=398.75
      For this problem we can use either of the methods shown above. Let's use Linear combination. Let's start by multiplying are first equation (x+ y=45) by -8.50.
              -8.50(x+ y=45)
             -8.50x -8.50y=-382.50
      Now we can add the two equations together!!
        -8.50x -8.50y=-382.50
            8.50x +9.75y=398.75
               1.25y=16.25
                   1.25     1.25
                       y=13
         Next we can plug "y" in to one of our equations and solve for "x"!!!
                    x+(13)=45
                       x=32
       Now let's check our work and will be done!!!
                (32)+(13)=45
                       45=45
       It's right!! Now we know that 32 pens were sold at $8.50 and 13 pens were sold at $9.75.

Now it's time to try it out on your own!!	Check your answers!!!
1. 8 small boxes plus 5 large boxes cost $184. A large box costs $3 more than a small box. What is the cost of each size box?	1. A large box costs $16, a small box costs $13.
2. Solution A is 2% alcohol. Solution B is 6% alcohol. 60 L of mixture are needed. How many liters of each solution are needed for the mixture to be 3.2% alcohol?	2. 42 L of 2% solution and 18 L of 6% solution.
3. The sum of a certain number and a second number is -42. The first number minus the second is 52. Find the numbers.	3. (5, -47)
4. One day a store sold 30 sweatshirts. White ones cost $9.95, and yellow ones cost $10.50. In all, $310.60 worth of sweatshirts were sold. How many of each color were sold?	4. 8 white shirts were sold and 22 yellow shirts were sold
5. Carlos is 8 years older than his sister Maria. Four years ago Maria was two thirds as old as Carlos. How old are they now?	5. Maria is 20 years old Carlos is 28 years old

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