Math 072 Chapter 3 Notes PDF

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This is a document of Chapter 3 lecture notes....

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3.1 - Systems of Equations of in Two Variables: Solutions by Graphing Pg1 Key ideas: • Be able to recognize if an ordered pair represents a solution to a system of equations. • Understand the idea of how to solve a system of equations by graphing.

Example 1. Consider the following statement. Can you find the solution? • Together, the Twins and the White Sox scored a total of 16 runs. • The Twins scored 6 more runs than the White Sox. • How many runs did each team score?

Solutions to systems of linear equations come in ordered pairs. 3 Ways to Solve Systems of Equations • Graphing (3.1)

• Substitution (3.2)

Example 2. Solve the following system of equations by graphing.

Math 072 Chapter 3 Notes Page 1

• Elimination (3.3)

3.1 - Systems of Equations of in Two Variables: Solutions by Graphing Pg2 Consistent vs Inconsistent Systems

Dependent vs, Independent Systems

Consistent System -- Has at least 1 solution.

Dependent -- Has a graph which contains the same line.

Inconsistent System -- Has no solutions.

Independent -- Has graphs with none of the same lines.

3 Possible Scenarios That Arise When Graphing 2 Lines.

Example 3. Identify each system as either consistent or inconsistent and then either dependent or independent. Also list the number of solutions for each system graphed. Scenario 1

Scenario 2

Scenario 3

Consistent or Inconsistent

Consistent or Inconsistent

Consistent or Inconsistent

Dependent or Independent

Dependent or Independent

Dependent or Independent

Number of Solutions?

Number of Solutions?

Number of Solutions?

Do you think that solving a system of equations via graphing is efficient? Explain why or why not.

Math 072 Chapter 3 Notes Page 2

3.2 - Systems of Equations - Substitution Key ideas: • Be able to solve a system of equations using the substitution method. • Recognize when to use the substitution method. • Understand how to determine if there are no solutions or infinitely many solutions.

Example 1.

Solve each system of Equations using the substitution method.

a.)

b.)

c.)

My Advice on When to use Substitution • Use when it is easy to solve for one variable. • Look for times when a variable is already by itself • When the coefficient of a variable is 1 and you prefer this method. Math 072 Chapter 3 Notes Page 3

3.3 Systems of Equations - Elimination Key ideas: • Be able to solve a system of equations using the elimination method. • Recognize when to use the elimination method. • Understand how to determine if there are no solutions or infinitely many solutions.

Example 1.

Solve each of the following equations using the elimination method.

a.)

b.)

c.)

d.)

Math 072 Chapter 3 Notes Page 4

3.4 - Applied Problems Systems of Equations Pg 1 Key ideas: • Recognize types of application problems involving systems of equations. • Be able to solve systems of equations.

Example 1. It takes Ashley 5 min to type a page of her essay and 1 ½ min to proofread a page. If she spends 2 hr typing or proofreading a total of 38 pages, how many pages did she type and proofread?

Example 2. Kevin’s Floral Emporium offers two types of sunflowers for sale by the stem. When in season, the small ones sell for $2.50 per stem, and the large ones sell for $3.95 per stem. One late summer weekend, Kevin sold a total of 118 stems for $376.20. How many of each size did he sell?

Interest Problem Example 3. William opened two investment accounts for his daughter’s college fund. The first year, these investments, which totaled $3200, yielded $155 in simple interest. Part of the money was invested at 5.5% and the rest at 4%. How much was invested at each rate?

Math 072 Chapter 3 Notes Page 5

3.4 - Applied Problems Systems of Equations Pg2 Mixture Problem Example 4. Soybean meal is 16% protein and corn meal is 9% protein. How many pounds of each should be mixed in order to get a 350-lb mixture that is 12% protein?

Distance Rate and Time Problem A Train leaves Danville Junction and travels north at a speed of 75 mph. Two hours later, a second train leaves on a parallel track and travels north at 125 mph. How far from the station will the second train catch up to the first train?

Math 072 Chapter 3 Notes Page 6

3.5 - Systems of Equations in 3 variables Key ideas: • Know the basic idea of how to solve a system of equations with 3 variables. Main Idea: Reduce 3x3 system to a more familiar 2x2 system! by eliminating the same variable from two equations. (Note: we just scratch the surface of 3x3 systems in this course.) Example 1.

Solve the system of equations.

Math 072 Chapter 3 Notes Page 7

3.7 - Systems of Inequalities Pg 1 Key ideas: • Know how to graph an inequality by identifying a test point and shading appropriately. • Understand how to graph a system of inequalities. Suppose the Vikings can score in one of two ways. • Touchdowns = 7 pts • Field Goals = 3 pts A sport analyst states that the Vikings will win their next game if they score at least 35 points. Write an inequality to represent this and construct a graph representing all possible solutions. Let T represent the number of touchdowns. Let F represent the number of field goals. Our system of inequalities might look like: Graph Representing the Solutions

On the graph to the right, the ordered pair is depicted. What does it represent?

Steps for Graphing Linear Inequalities

Step 1 - Replace Inequality with an equals sign for graphing purposes. Step 2 • Graph the line with a dashed line if the inequality is < or >. • Graph the line with a solid line if the inequality is or Step 3 - Pick a test point NOT on the line. Substitute it into the original inequality. • If True -------- Shade the same side as the test point. • If False -------- Shade the opposite side as the test point.

Math 072 Chapter 3 Notes Page 8

3.7 - Systems of Inequalities Pg 2 Example 1. Graph each of the following inequalities

Step 1

Step 2 Dashed or Solid Line? Step 3 Test Point:

Example 2. Graph each of the following inequalities

Step 1

Step 2 Dashed or Solid Line? Step 3 Test Point:

Math 072 Chapter 3 Notes Page 9

3.7 - Systems of Inequalities Pg 3 Example 3. Graph the following System of Inequalities

Example 4. Graph the following System of Inequalities

Inequality

Test Point

True/False

Math 072 Chapter 3 Notes Page 10

Shading

3.7 - Systems of Inequalities Pg 4 Example 5.

Inequality

Graph the following System of Inequalities

Test Point

True/False

Math 072 Chapter 3 Notes Page 11

Shading...