Upcoming SlideShare
×

# 05 systems of equations in two variables

2,841 views

Published on

Published in: Technology
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
2,841
On SlideShare
0
From Embeds
0
Number of Embeds
111
Actions
Shares
0
196
0
Likes
0
Embeds 0
No embeds

No notes for slide

### 05 systems of equations in two variables

1. 1. Systems of Linear Equations in Two Variables<br />
2. 2. Let’s try this!<br />For each pair of equations,<br />Sketch the graph<br />Describe the relationship between the lines on the graph<br />
3. 3. Systems of Linear Equations in Two Variables<br />
4. 4. System of Linear Equations is a set of two or more linear equations, which are to be treated simultaneously, generally to solve for values of the variables that satisfy all of the equations, if there are such values.<br />
5. 5. If a system has a solution, it is called consistent; if it has no solution, it is inconsistent.<br />If a system is made up of 2 equivalent equations (coincide), such system is called a dependent system; otherwise, it is independent.<br />
6. 6. In general, given the system of linear equations<br />If , <br />the system is consistent and independent<br />graph: 2 lines intersect at exactly one point<br />Unique solution (pt. of intersection)<br />
7. 7. In general, given the system of linear equations<br />If , <br />the system is inconsistent<br />Graph: Two distinct parallel lines.<br />No solution <br />
8. 8. In general, given the system of linear equations<br />If , <br />the system is dependent<br />graph: Two lines coincide and are actually the same line<br />infinitely many solutions; every solution of either equation is a solution of the other<br />
9. 9. Solving Systems of Equation:<br />1. Graphical Solution<br />2. Algebraic Solution<br />2.1 Elimination Method<br />2.2 Substitution Method<br />3. Cramer’s Rule<br />
10. 10. Graphical Solution<br />Use <br />Geometer’s Sketchpad<br />Wzgrapher<br />Graphing calculator<br />
11. 11. Calculator Key Strokes <br />MODE<br />Choose <br />5: EQN<br />1:anX + bnY = cn<br />Input values<br />Ex. x + y = 3<br /> x – y = 3<br />a1 = 1 b1 = 1 c1 = 3<br />a2 = 1 b2 = -1 c2 = 3 <br />Press equal sign twice<br />X = 3, Y = 0<br />
12. 12. ELIMINATION METHOD<br />Eliminate one variable by addition or subtraction of the equations and then solve for the solution of the remaining variable.<br />
13. 13. Example: <br />Find the solution to the given system of linear equations.<br />
14. 14. Example: <br />Solution:<br />Write both equations in the same form (Ax +By =C).<br />Multiply one or both of the equations by appropriate numbers (if necessary) so that one of the variables will be eliminated by addition.<br />
15. 15. Example: <br />Solution:<br />Add the equations to get an equation in one variable.<br />Solve the equation in one variable.<br />
16. 16. Example: <br />Solution:<br />Substitute the value obtained for one variable into one of the original equations to obtain the value of the other variable.<br />
17. 17. Example: <br />Solution:<br />Check the two values in both of the original equations.<br />
18. 18. Example: <br />Find the solution to the given system of linear equations.<br />
19. 19. Example: <br />Solution:<br />Write both equations in the same form (Ax +By =C).<br />Multiply one or both of the equations by appropriate numbers (if necessary) so that one of the variables will be eliminated by addition.<br />
20. 20. Example: <br />Solution:<br />Add the equations to get an equation in one variable.<br />Solve the equation in one variable.<br />
21. 21. Example: <br />Solution:<br />Substitute the value obtained for one variable into one of the original equations to obtain the value of the other variable.<br />
22. 22. Example: <br />Solution:<br />Substitute the value obtained for one variable into one of the original equations to obtain the value of the other variable.<br />
23. 23. Example: <br />Solution:<br />Check the two values in both of the original equations.<br />
24. 24. Exercises:<br />Find the solution to the given systems of linear equations.<br />
25. 25. SUBSTITUTION METHOD<br />Solve one equation for one of the variables and substitute this expression into the other equation. Then solve for the variable. <br />It is easiest to use the method of substitution when one of the coefficients in an equation is 1.<br />
26. 26. Example:<br />Find the solution to the given system of linear equations.<br />Solution:<br />Solve one of the equations for one variable in terms of the other. Choose the equation that is easiest to solve for x or y.<br />
27. 27. Example:<br />Find the solution to the given system of linear equations.<br />Solution:<br />Substitute into the other equation to get an equation in one variable.<br />
28. 28. Example:<br />Find the solution to the given system of linear equations.<br />Solution:<br />Solve for the remaining variable (if possible).<br />
29. 29. Example:<br />Find the solution to the given system of linear equations.<br />Solution:<br />Substitute the value just found into one of the original equations to find the value of the other variable.<br />
30. 30. Example: <br />Solution:<br />Check the two values in both of the original equations.<br />
31. 31. Exercises:<br />Find the solution to the given systems of linear equations.<br />