1. Systems of Linear Equations
2.4
SECTION
Systems of Linear Equations
Copyright © Cengage Learning. All rights reserved.

2. Learning Objectives
1 Determine the solution to a system of equations algebraically, graphically, and using technology and interpret the real-world meaning of the results
2 Use the substitution and elimination methods to solve linear systems that model real-world scenarios
3 Determine if systems of linear equations are dependent or inconsistent and explain the real-world meaning of the results

3. Systems of Linear Equations

4. Systems of Linear Equations
A system of equations is a group of two or more equations. To solve a system of equations means to find values for the variables that satisfy all of the equations in the system.
Systems of equations can involve any number of equations and variables; however, we will limit ourselves to situations containing two variables in this section.

5. Systems of Linear Equations
A solution of a pair of linear equations is an ordered pair of numbers that satisfies both equations. The ordered pair (5, 3) is a solution of the linear equations below.
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 5

6. Systems of Linear Equations
Although (7, 2) makes the first equation true in the system…
(2)(7) + (4)(2) = 22
…it does not make the second equation true.
7 – (6)(2)  5
Therefore, it is not a solution for the system.
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 6

7. Systems of Linear Equations
Graphing a system of two linear equations in two unknowns gives one of three possible situations:
Case 1: Lines intersecting in a single point. The ordered pair that represents this point is the unique solution for the system.
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 7

8. Systems of Linear Equations
Case 2: Lines that are distinct parallel lines and therefore don’t intersect at all. Because the lines have no common points, this means that the system has no solutions.
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 8

9. Systems of Linear Equations
Case 3: Two lines that are the same line. The lines have an infinite number of points in common, so the system will have an infinite number of solutions.
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 9

10. Solving Systems Using the Elimination Method
Example: Solve the system.
(continued on next slide)
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 10

11. Solving Systems Using the Elimination Method
Example: Solve the system.
Solution:
Multiply the top equation by 3 and the bottom equation by 2 to get opposite coefficients for y.
(continued on next slide)
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 11

12. Solving Systems Using the Elimination Method
Next add corresponding sides of both equations and the y drops out.
Solve for x.
(continued on next slide)
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 12

13. Solving Systems Using the Elimination Method
To find y, substitute 2 for x in either equation of the original system.
Thus, the solution for this system is (2, 1) (Case 1).
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 13

14. Solving Systems Using the Elimination Method
Example: Solve the system
(continued on next slide)
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 14

15. Solving Systems Using the Elimination Method
Example: Solve the system
Solution:
Multiply both sides of the top equation by 4 to clear the fractions.
(continued on next slide)
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 15

16. Solving Systems Using the Elimination Method
Multiply the bottom equation by 2 and add the equations to eliminate x from the system:
There are no points common to both lines (Case 2).
A system that has no solutions is said to be inconsistent.
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 16

17. Solving Systems Using the Elimination Method
Example: Solve the system
(continued on next slide)
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 17

18. Solving Systems Using the Elimination Method
Example: Solve the system
Solution:
Multiply the top equation by 10 and the bottom equation by 100 to get rid of the decimals.
(continued on next slide)
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 18

19. Solving Systems Using the Elimination Method
Multiply the top equation by –5, and add the two equations to eliminate x from the system.
The two lines must be the same (Case 3).
A system that has an infinite number of solutions is said to be dependent.
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 19

20. Solving Systems Using the Elimination Method
© 2010 Pearson Education, Inc. All rights reserved.
Section 8.1, Slide 20