Systems of Linear Equations (2 variables) by xab70192

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									Systems of Equations
To solve a non-square system of equations (number of variables does not equal the number of equations):
         A. 3 equations and 2 unknowns (usually inconsistent or independent).
                  1. Put each equation in slope-intercept form and graph.
                  2. Draw graph to show your work.
                  3. If all three lines intersect at the same point, that coordinate is the solution (independent). Use the “calculate: intersect”
                  function to find coordinate.
                  4. If the three lines do not intersect at the same point, there is no solution (inconsistent).
         B. 2 equations and 3 unknowns (usually dependent system)
                  1. Use elimination with the 2 original equations to get rid of “x”.
                  2. Solve for “y”.
                  3. Use elimination with the 2 original equations to get rid of “y”.
                  4. Solve for “x”.
                  5. Write the solution (x,y,z) with “z” as the dependent variable.




   2 x  4 y  3                                  3 x  5 y  22
                                                                                               x  2 y  z  8
1.  x  y  7                                 2.  x  2 y  8                               3. 
   3 x  2 y  8                                 3 x  4 y  4                                 3x  7 y  6 z  26
                                                 




    x  2 y  3z  9                             x  2 y  z  2                                x  2 y  6
4.                                            5.                                            6. 
   2 x  5 y  5 z  17                          2 x  y  z  1                               3x  3 y  9
   2 x  3 y  7             2 x  4 y  12             2 x  2 y  12
7.                         8.                         9. 
   3x  y  5                 x  2 y  6             x  y  6




    2 x  y  5                3x  2 y  12              3x  2 y  18
10.                        11.                        12. 
    3x  2 y  4               2 x  4 y  8               3x  2 y  12




    2 x  y  z  13            x  2 y  3z  9          4 x  y  5 z  11
                                                          
13.  x  2 y  z  2       14.  x  3 y  4         15.  x  2 y  z  5
    8 x  3 y  4 z  2       2 x  5 y  5 z  17       5 x  8 y  13z  7
                                                          
    3 x  2 y  4 z  11                                       x  2 y  7 z  4
                                                              
16. 2 x  2 y  4 z  6                                   17. 2 x  y  z  13
    2 x  3 y  6 z  8                                       3 x  9 y  36 z  33
                                                              




Systems of Quadratic Equations

To solve a system of equations when at least one of the equations is a quadratic, you can use one of several methods.
Elimination, substitution, and graphing will all work and give you the same answer. To use elimination, multiply one or
both of the equations by a constant so that a variable will cancel out. Then solve the remaining equation. To use
substitution, solve for the same variable on both equations and then set them equal and solve the remaining equation. To
use graphing, solve both equations for y, use your calculator to graph, and then find the intersection points.

        Ex 1  x  y  4
                2
               x  y  3




        Ex 2       x 2  x  y  1
                   
                    x  y  1




        Ex 3
                 3 x  4 y  18
                  2
                 x  y  36
                          2
Ex 4
       R y  1
       |x
       Sx  y  4
                 2


       |3
       T             2




Ex 5
       R y
       |x2

       Sy
                     2
                         1
       |x
       T 2           2
                         4




Ex 6
       Rx  5y  9
       2
       S  y  25
       Tx2           2




Ex 7
       Rx  y  5
       |
       Sy3
             2


       |x
       T 2




Ex 8
       R y  1
       x
       S y  1
                 2


       T
       x
Systems of Linear Inequalities

Graph each inequality. Remember to use solid or dotted lines appropriately. Shade the region where the inequalities
overlap. Find the points of intersection and label them on the graph. If the region is not bounded on all sides, the region
is called an infinite wedge.

Ex 1    5 x  3 y  6
        
        5 x  3 y  9




Ex 2    x  y  2
        
        x  2
        y  3
        




Ex 3    2 x  y  2
        
        x  2
        y  1
        
   Rx  3y  6
   |2 x  y  2
4. S2
   |x  2
   T




   R 4
   |x  3
5. Sy
   |y  2  x
   T




   R 2 x  4
    y
   |2 y  x  4
6. S
   |5x  3y  15
   T
Systems of Quadratic Inequalities

Graph each inequality. Remember to use solid or dotted lines appropriately. Shade the region where the inequalities
overlap. Then solve the system using substitution. Use the same method used for Quadratic Equations. Label the
intersection points on the graph.

Ex 1    x 2  y  1
        
         x  y  1




        x  4 y  y2
Ex 2    
        y  x




Ex 3   x 2  y  5
       
       2 x  4 y  0




Ex 4
       x 2  y  9
       
       x  y  5
Ex 5
        R  6y  x
        y   2

        Sy  3x
        T
        




     y2  2y  x
Ex 6 
     3x  2 y  4




      y 2  8 y  12   x
Ex 7 
      y  2x




      x  26  y 2  10 y
Ex 8 
      x  2




     x  6 y  y 2
Ex 9 
     y  1

								
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