Solving Systems of Two Equations in Two Variables

Intermediate Algebra - Chapter 3 Summary

Sec 3-1 & 3-2: SOLVING SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES

1. What does it mean to solve a system of two equations in two variables?

If a system of equations has at least one solution, it is ______. If a system of equations has exactly one solution, it is ______. If a system of equations has an infinite number of solutions, it is ______.

2. METHODS FOR SOLVING A SYSTEM OF EQUATION: SOLVE BY GRAPHING 6x – 2y = 7 2x + y = 4

Graph the equations on the same coordinate plane. The point of intersection represents the solution. Check the solution by substituting the coordinates into each equation.

Solution: ______Classify this system of equations:

SOLVE BY SUBSTITUTION SOLVE BY ELIMINATION

Intermediate Algebra – Ch. 3_Summary pg 1 of 7 SOLVE BY GRAPHING 3x – 2y = 6 6x – 4y = 12

SOLVE BY GRAPHING

Intermediate Algebra – Ch. 3_Summary pg 2 of 7 3. Applications: a) A lab technician is mixing a 10% saline solution with 4% saline solution. How much of each solution is needed to make 500 milliliters of a 6% saline solution?

b) You are choosing between two cell-phone plans. Plan A charges a flat fee of $50 per month for unlimited calls. Plan B charges a monthly fee of $20 plus $0.05 per minute. • Express the monthly cost, A, for plan A as a function the number of minutes of calls, x. • Express the monthly cost, B, for plan B as a function the number of minutes of calls, x. • For how many minutes of calls will the costs of the two plans be the same?

c) Josh and his band are planning to record their first CD. The initial start-up cost is $1,500, and each CD will cost $4 to produce. They plan to sell their CDs for $10 each. How many CDs must the band sell before they make a profit? (In businesses, the point at which the income equals the cost is called the break-even point.)

Intermediate Algebra – Ch. 3_Summary pg 3 of 7 Sec 3-3 & 3-4: SOLVING SYSTEMS OF LINEAR INEQUALITIES IN TWO VARIABLES BY GRAPHING

1. Daryl wants to buy some books. Each book costs $8. He can spend no more than $90 on books. Find all possible number of books that he can buy.

2. Daryl wants to buy some books and CDs. A book costs $8 and a CD costs $15. He can spend no more than $90 on books and CDs. Find all possible combinations of books and CDs that he can buy.

3. Find all possible pairs of coordinates that satisfy each inequality or system of inequalities. a) 2x + y ≥ 0.

b) y > -2x + 4 y ≤ 2

Intermediate Algebra – Ch. 3_Summary pg 4 of 7 c. y > x + 1 x ≤ 3

d. x + y ≥ -1 x – y ≤ 6 y ≤ 4 Find the coordinates of the vertices formed by the system of inequalities. Find the area of the region.

5. The available parking area of a parking lot is 600 square meters. A car requires 6 square meters of space, and a bus requires 30 square meters of space. The attendant can handle no more than 60 vehicles. Set up and graph a system of linear inequalities to show all possible combinations of cars and buses that meet the constraints.

Intermediate Algebra – Ch. 3_Summary pg 5 of 7 Sec 3-5: LINEAR PROGRAMMING

1. A carpenter makes bookcases in two sizes, large and small. It takes 6 hours to make a large bookcase and 2 hours to make a small one. The profit on a large bookcase is $50, and the profit on a small bookcase is $20. The carpenter can spend only 24 hours per week making bookcases and must make at least 2 of each size per week. Find the maximum possible profit in this situation.

What is linear programming? A method for solving problems in which a particular quantity that must be maximized or minimized is limited by other factors. Procedure for solving a linear programming problem: 1. Define the variables. 2. Write a system of inequalities that represents the constraints. 3. Graph the system of inequalities. 4. Find the coordinates of the corner points (vertices of the feasible region). 5. Write the objective function (to be maximized or minimized.) 6. Find the value of the objective function at each corner point of the graphed region. 7. Use the results in step 6 to determine the maximum (or minimum) value of the objective function.

More Practice:

2. I am thinking of a point with coordinates (x, y) in the coordinate plane that makes the quantity 3x + y as large as possible. The ordered pair has to meet all of the following conditions: y ≥ 1 x ≤ 6 y ≤ 2x + 1 What choice(s) of (x, y) would work?

Intermediate Algebra – Ch. 3_Summary pg 6 of 7 3. A television manufacturer makes console and wide-screen televisions. It is bound by the following constraints: • Equipment in the factory allows for making at most 450 console televisions and at most 200 wide-screen televisions in one month. • The cost to the manufacturer per unit is $600 for the console TVs and $900 for the wide-screen TVs. Total monthly costs cannot exceed $360,000. The profit per unit is $125 for the console TVs and $200 for the wide-screen TVs. How many of each type of televisions should be made to maximize the profit?

4. A school is preparing lunch menus containing foods A and B. The specifications for the two foods are given in the following table:

Food Units of fat per Units of Carbohydrates per Units of protein per ounce ounce ounce A 1 2 1 B 1 1 1

Each lunch must provide at least 6 units of fat per serving, no more than 7 units of protein, and at least 10 units of carbohydrates. The school can purchase food A for $0.12 per ounce and food B for $0.08 per ounce. How many ounces of each food should a serving contain to meet the dietary requirements at the least cost?

Intermediate Algebra – Ch. 3_Summary pg 7 of 7