1. Systems of Linear Equations MATH 102 Contemporary Math S. Rook

2. Overview Section 8.1 in the textbook: – Solution to a system of equations – Solving a system of equations by elimination – Modeling with a system of equations

3. Solution to a System of Equations

4. Systems of Equations in General System of Equations: examining two or more equations simultaneously – For our purposes the systems of equations we will study will be linear (e.g. ) We are often concerned with the solution to a system of equations 4

5. Verifying the Solution to a System of Equations What do you think the solution to is whose graph is represented on the right? – A solution to a system of equations is a point (i.e. has an x-coordinate and a y-coordinate) Using the given system, how can we verify that (1, 3) is the solution? – i.e. How would we verify that (1, 3) lays on the line 2x + y = 5? For a point to be a solution to a system of equations, it MUST satisfy BOTH equations! 5

6. Verifying the Solution to a System of Equations (Example) Ex 1: Show whether or not the points satisfy the given system of equations: ; (-5, 2), (-2, 0) 6

7. Solving a System of Equations by Elimination

8. Solving a System of Equations Using Elimination Consider solving First, we choose one of the variables to eliminate To eliminate the variable, we add the two equations together – What happens when we add the two equations together? What do you notice about the coefficients in front of y before the elimination step? – To eliminate a variable, the coefficients in front of the variable MUST be opposites 8

9. Solving a System of Equations Using Elimination (Continued) Now consider solving – Will one of the variables be eliminated if we add the equations together? Which variable should we eliminate? (note that there are no wrong choices) What must be true about the coefficients in front of the variable that we wish to eliminate? Use the concept of the LCM to find the opposites We cannot multiply just one part of an equation by a number – Multiply the ENTIRE equation by the constant 9

10. Solving a System of Equations by Elimination (Example) Ex 2: Solve the system of equations by elimination: a)c) b)d)

11. 11 Inconsistent & Dependent Systems Inconsistent system: a system of equations which has NO solution – Graphically, the lines will never intersect – In other words, the lines are parallel Dependent system: a system of equations which has an INFINITE number of solutions. – Graphically, the lines lie on top of each other – “All real numbers” DOES NOT apply Use the term “Infinite Solutions”

12. Inconsistent & Dependent Systems of Equations (Continued) If BOTH variables are eliminated, we are dealing with an inconsistent or dependent system – Must determine whether the resulting statement is true or false – Remember that the solution for a TRUE statement is NOT all real numbers!!! 12

13. Solving a System of Equations by Elimination (Example) Ex 3: Solve the system of equations by elimination: a) b)

14. Modeling with a System of Equations

15. Objective is to extract a system of equations from a word problem: – Be on the lookout for key words to help set up each equation – Each equation represents a component of the word problem Identifying these components makes the problem easier to understand Use elimination to solve the resulting system of equations MUST practice to become proficient!!! 15

16. Modeling with a System of Equations (Example) Ex 4: Solve: Peanuts worth $2.50 per pound and cashews worth $3.25 per pound are used to make 15 pounds of a trail mix worth $3.00 per pound. If only peanuts and cashews are used, how many pounds of each nut are present in the trail mix? 16

17. Modeling with a System of Equations (Example) Ex 5: Solve: At a certain baseball stadium, one can either sit in field level seats or upper level seats. If tickets are purchased for 12 upper level seats and 8 field level seats, one pays $456. If tickets are purchased for 6 upper level seats and 16 field level seats, one pays $588. How much does each type of ticket cost? 17

18. Summary After studying these slides, you should know how to do the following: – Verify whether a point satisfies a system of equations – Solve a system of equations by elimination – Model and solve a problem using a system of equations Additional Practice: – See problems in Section 8.1 Next Lesson: – Proportions & Variation (Section 7.5)