EASY MATH
              System of equation

 
 
A project to Mrs. 
Felz class of Algebra II  to teach how to work a problem that can be difficult to work and we can make it easier to learn.
    Word Problem        
A total of $ 27,000 was invested , part of it at 10% and part at 12%. The total yield was $2990. How much was invested at each rate? 
 

 Remember - Problem - solving Guidelines
    1. Understand the problem
    2. Develop and carry out a Plan
    3. Find the Answer and Check

 1.The total is $ 27,000 invested 2.parts at 10% and 12 %    3. The equation is x + y = $ 27,000 4.The other equation is   10% x + 12% y = $ 2990  5. Now you use subtitution to solve it.
                          x + y = 27 000
                   10% x +12% y = 2990
         1. You don't want to confuse your-self, so we change the percent into decimal to be more easier.
                       x + y = 27 000
                    .10 x + .12 y = 2990 
         2. Now you work it.
               - .10 (x + y = 27000) 
                     .10x + .12y = 2990

                   - .10x + -.10 y =  -2700
                    .10x + .12y = 2990 
                      .02y/.02 = 290/.02
                                y = 1450


                            -.12 (  x + y = 27000)
                             .10x + .12y = 2990
   -.12x + -.12 y = -3240
    .10 x + .12 y = 2990
     -.02x /-.02   = -250/ -.02
                    x = 12500

    3.Then you got your two answers. And you are done.
 


 

 
         NEW PROBLEM   (Linear Convination)
     0.3x + 0.2y = 0.3
     0.2x + 0.3y = -0.3		    1. You deside what letter you are going to remove.
         I will remove ''x''.

                0.3x + 0.2y = 0.3
             0.2x + 0.3y =-0.3

    2. The chose what number you are going to use to cancel 
        both x's. ( I will chose 2 and 3 to cancel the x's.)

                2(0.3x + 0.2y = 0.3)           (Remember that one
                (0.2x + 0.3y =-0.3)-3               has to be negative.)
                 0.6x + 0.4y = 0.6
              -0.6x -0.9y = 0.9
                        -0.5y = 1.5
                           -0.5      -0.5<
                            y = -3    -------> > You are done only for y , now find x by doing the same as the y.
        3. Now find x.

           3 (0.3x + 0.2y = 0.3)
                (0.2x + 0.3y =-0.3)-2
                 0.9x + 0.6y = 0.9
              -0.4x -0.6y = 0.6
              0.5x          = 1.5
                 0.5                  0.5
                         x = 3   ----------> >   You have x now and you got your two points.

        4. (3,-3)  You are done.
 


 

 
        New section ( Substitution)

                          ( y = 6 - 2 x) 

                  
                2x + y = 6
                 3x + ay = 4
 

 1. You solve the first equation for y because its y- term has a coefficient of 1.
                          y = 6 - 2x

2. You can substitute 6 - 2x for y in the socond equation.

                    3x + 4y = 4
                   3x + 4(6 - 2x) = 4 >>>>>> Substituring 6 - 2x

  2. This gives you an equation in one varisble. Then yoy can solve for x.

                3x + 24 - 8x = 4 >>>>>>>Using the distributive
                            -5x = -20               property.
                               -5     -5
                                x = 4 
3. Now you can subtitude 4 for x in either equation and solve for y.

                   2x + y = 6 
                 2 (4) + y = 6
                            y = -2
4. Then you obtain (4, -2). 
 
 

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