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

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

Systems of Linear Equations

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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