SCC Ch 3. Math 124 Intermediate Algebra: Part II Sec 3.1: Systems of Linear Equations in Two Variables

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1 Math 124 Intermediate Algebra: Part II Sec 3.1: Systems of Linear Equations in Two Variables Overview of Math 124 Chapter 3: Systems of Linear Equations and Inequalities (omit 3.4, 3.5) Chapter 7: Quadratic Equations and Functions Chapter 8: Exponential and Logarithmic Functions Chapter 9: Conics (omit 9.4, 9.5) Chapter 10: Sequences, Series, and the Binomial Theorem A system of equations is a set of 2 or more equations, in 2 or more variables, for which a common solution is sought. Graphs intersect at one point. Graphs are parallel. Equations have the same graph. One Solution No Solution Infinite Number of Solutions... Consistent Inconsistent Consistent Independent Independent Dependent Defns consistent system: at least one solution dependent system (for 2x2): one equation is a constant multiple of the other Ex 1 Determine whether the given ordered pairs are solutions to the system of equations. a) b) Ex 2 Solve the system by graphing. Ex 3 Solve the system by substitution. Page 1 of 8

2 Ex 4 Solve the system by elimination. Ex 5 Solve each using the method of your choice. If there is not a unique solution, provide the reason. When should we use the graphical method? Only when a visual solution is sought. a) b) c) Sec 3.2: Problem Solving: Systems of Two Linear Equations Containing Two Unknowns Ex 1 (# 12) Juan is thinking of two numbers. He says that 3 times the first number minus the second number is 118. In addition, two times the first number plus the second number is 147. Find the numbers. Define variables and set up only. Page 2 of 8

3 Ex 2 (# 14) Johnny has $6.75 in dimes and quarters. He has 8 more dimes than quarters. How many quarters does Johnny have? How many dimes does Johnny have? Ex 3 (# 20) Marge and Homer have $80,000 to invest. Their financial advisor has recommended that they diversify by placing some of the money in stocks and some in bonds. Based upon current market conditions, he has recommended that three times the amount in bonds should equal two times the amount invested in stocks. How much should be invested in stocks? How much should be invested in bonds? Ex 4 (# 22) A candy store sells chocolate-covered almonds for $6.50/lb. and chocolate-covered peanuts for $4.00/lb. The manager decides to make a bridge mix that combines the almonds and peanuts. She wants the bridge mix to sell for $6.00/lb., and there should be no loss in revenue when selling the bridge mix versus the almonds and peanuts alone. How many pounds of chocolate-covered almonds and chocolate-covered peanuts are required to create 50 pounds of bridge mix? Page 3 of 8

4 Ex 5 (# 24) A Piper Arrow can fly 510 miles in 3 hours with a tailwind. Against this same wind, the plane can fly 390 miles in 3 hours. Find the airspeed of the plane (speed of plane in still air). What is the impact of the wind on the plane? Ex 6 (# 34) The sum of four times a first number and a second number is 68. If the first number is decreased by twice the second number, the result is -1. Find the numbers. Ex 7 (# 40) A doctor s prescription calls for the creation of pills that contain 10 units of vitamin and 13 units of vitamin. Your pharmacy stocks two powders that can be used to make these pills: Powder contains 20% vitamin and 40% vitamin ; Powder contains 50% vitamin and 30% vitamin. How many units of each powder should be mixed in each pill? Page 4 of 8

5 Sec 3.3: Systems of Linear Equations in Three Variables Defn A linear equation in three variables can be written in the form, where and are real numbers (not all 0). The aforementioned equation is the standard form of a linear equation in 3 variables. A solution of a system of 3 equations in 3 variables is an ordered. Refer to page 256 (in Sullivan) for figures that illustrate some possibilities geometrically. exactly one solution planes intersect at one point infinite number of solutions planes intersect along a common line no solution three parallel planes or two planes intersect at a time (with no point common to all 3) Note: Must (1) eliminate same variable (twice) and (2) use all equations. Ex 1 Determine which of the following ordered triples are solutions to the system of equations. a) b) Ex 2 Solve each system. a) (# 18) b) (# 22) Page 5 of 8

6 c) (# 40) d) (# 20) e) (# 28) Practice Problem Systems of Linear Inequalities Ex 1 (# 12) Determine which of the points, if any, satisfy the system. a) b) Page 6 of 8

7 Ex 2 Graph each system of linear inequalities. a) (# 22) b) (# 26) Ex 3 (# 30) Graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points. Page 7 of 8

8 Ex 4 (# 38) Bob s Printer Emporium manufactures two printers, a color ink jet printer, and a black-andwhite laser printer. Each color printer requires 2 hours for molding and 3 hours for assembly: each laser printer requires 3 hours for molding and 4 hours for assembly. There are a total of 80 hours per week available in the work schedule for molding and 120 hours per week available for assembly. a) Using to denote the number of color printers and to denote the number of laser printers, write a system of linear inequalities that describes the possible number of each model printer that can be manufactured in a week (it is possible to manufacture a portion of a printer). b) Graph the system and label the corner points. Page 8 of 8