Note: Two perpendicular lines form a system of the first type. (Nothing special about being )

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1 Math 100 Elementary Algebra Sec 7.1: Solving Linear Systems by Graphing Two or more equations [inequalities] in several variables that are considered simultaneously are called a system of equations [inequalities]. What are the (graphical) possibilities of a system of equations consisting of two linear equations in two variables? Lines Intersect Lines are Parallel (Do not intersect) Lines Coincide (Intersect at every point on the line) One Solution No Solution Infinitely many solutions (x, y) } Note: Two perpendicular lines form a system of the first type. (Nothing special about being ) A system of linear equations that has no solutions is called inconsistent. Otherwise, it is consistent. A system of linear equations that has an infinite number of solutions is called dependent. Otherwise, it is independent. Ex 1 Solve by graphing. If there isn t a unique solution to the system, state the reason (book and test answer). Use correct notation in answer! Solve for y for slope but don t use y-intercept. a) b) 2x 4y = 8 2x y = 1 2x + 3y = 3 3x + y = 21 Page 1 of 12

2 c) d) 3x 6y = 6 x 2y = 4 x y = 3 2x 2y = 6 e) Challenge PP f) Challenge PP 4x 6y = 8 2x + 3y = 4 3x + 2y = x + y = 5 2 Sec 7.2: The Substitution Method 4x 5y = 22 Ex 2 Is (3, 2) a solution to the system x + 2y = 1? What does this mean graphically? Page 2 of 12

3 Ex 3 Solve the system. If there is not a unique solution, state the reason (textbook vs exam answer). Be sure to use correct notation! Also, provide the graphical interpretation. a) b) 2 x + 4y = 22 5x + 3y = x y = 1 5 2x y = 1 c) d) CHECK your answer. 1 x 2y = 1 3 x 6y = 3 3x + 9y = 27 x + 3y = 1 Must show all steps. Check for part (d) e) Start/rearrange. Finish as PP. 2(2x y 11) = 1 y 3(3x + y) 17 = 2(4x + y) Answer: (8, 9) Page 3 of 12

4 Sec 7.3: The Elimination Method Ex 4 Solve using the Elimination (Addition) Method. Check answers. a) b) x + y = 2 x + 3y = 4 12x + 5y = 9 2x 7y = 25 Ex 5 What variable would you eliminate and how would you do so? a) b) c) 2x + 5y = 3 5x + 3y = 23 5x + 4y = 5 7x + 6y = 36 12x 24y = 36 14x + 2y = 7 Ex 6 a) x 4 y 2 = 1 x 3 y = 2 Solve each system. Page 4 of 12

5 b) Prac Prob c) 4 7 x + 3 y = x + y = 19 5 Answer: ( 4, 3) 7 + 3(a + b) = b + 8 3(b 4) = 2a 3 Answer: ( 3, 5) Ex 7 Solve each system using the method of your choice. If there is not a unique solution, state the reason (book vs exam answer). Be sure to use correct notation! Also, provide the graphical interpretation. a) b) 0.05x y = x 2y = 8 0.1x y = x + 3y = 12 c) Prac Prob d) Prac Prob 4x + 3y = 0 5x 6y = 9 Answer: ( 3, 4) 5x 3y = x + 0.4y = 30 Answer: (20, 30) Page 5 of 12

6 e) f) 15x 18y = 10 x y = x + 3y = 7 2 y x = g) h) 1 (x + y) = (x y) = x y = x y = 1 2 Page 6 of 12

7 When should elim/subst be used? i) Prac Prob j) x 6y = 26 3x 7y = 34 Ans: ( 2, 4) x + 2y 1 = 3y + 5 7x = 6y + 42 k) l) y = 3 4 x + 7 x = 2y 2 y = 1 2 x + 2 3(x 2y + 1) = 15 Page 7 of 12

8 Ex 8 (#6) Sec 7.4: Applications of Systems of Equations One number is 2 more than 3 times another. Their sum is 26. Find the two numbers. Solve PP Ans: 6 and 20 Ex 9 (#8) The difference of two positive numbers is 8. The larger number is twice the smaller decreased by 7. Find the two numbers. Page 8 of 12

9 Ex 10 (#12) Michael is scheduled to receive an inheritance when he is 6 more than a third the age of his grandfather. The difference between their ages will be 38 years. Find their ages at the time of inheritance. Solve PP Ans: 28 and 66 years old Ex 11 (#14) A total of $11,000 was invested. Part of the $11,000 was invested at 4%, and the rest invested at 7%. If the investments earn $680 per year, how much was invested at each rate? Page 9 of 12

10 Ex 12 (#20) A coin collector has 31 dimes and nickels with a total face value of $2.40. (They are actually worth a lot more.) How many of each coin does she have? Ex 13 (#24) How much 50% antifreeze solution and 40% antifreeze solution should be combined to give 50 gallons of 46% antifreeze solution? Page 10 of 12

11 Ex 14 (#30) Tyler has been saving his winning lottery tickets. He has 23 tickets that are worth a total of $175. If each ticket is worth either $5 or $10, how many of each does he have? Solve PP Ans: 11 $5 tickets, 12 $10 tickets 2x 3y = 6 Ex 15 Solve 11x 3y = 6 Ex 16 Melissa drove to Dallas while Kerri drove to Houston in the same amount of time. Melissa drove 360 kilometers, while Kerri drove 280 kilometers. Melissa traveled 20 kilometers per hour faster than Kerri on her trip. What was the (average) speed in kilometers per hour for each woman? Solve using two variables. Page 11 of 12

12 Ex 17 Practice Prob A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with 4% cement should he mix with 3 bags of 8% cement to produce a mix containing 5% cement? Answer: 9 bags Summary: What should you expect when checking answers in each of the following cases? Ex 1: The answer is ( 2, 0). Ex 2: The answer is. Ex 3: The answer is all ordered pairs on the line (infinite solutions), written in this form: (x, y) }. Page 12 of 12

Final Exam Practice Problems Simplify completely.

1) Final Exam Practice Problems Simplify completely. d) e) (Decimal answer ok here) f) g) 2) 3) d) 4) Do NOT leave an exponent in your answer for a)-c). Write final answer with positive exponents. d) e)