Section 8.1 Solving Systems of Linear Equations by Graphing

OBJECTIVES Solve a system of two equations in two variables by graphing. A

OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent. B

OBJECTIVES Solve an application. C

SOLVING SYSTEMS OF SIMULTANEOUS EQUATIONS There are three possible solutions.

POSSIBLE SOLUTION 1 Consistent and independent equations: The graphs of the equations intersect at one point, whose coordinates give the solution of the system.

POSSIBLE SOLUTION 2 Inconsistent equations: The graphs of the equations are parallel lines; there is no solution for the system

POSSIBLE SOLUTION 3 Dependent equations: The graphs of the equations coincide (are the same). There are infinitely many solutions for the system.

COMPARISONS Intersecting Lines Have different slopes

COMPARISONS Intersecting Lines Have one solution

COMPARISONS Intersecting Lines Form a consistent system

COMPARISONS Parallel Lines Have the same slopes

COMPARISONS Parallel Lines Have different y-intercepts

COMPARISONS Parallel Lines Have no solution

COMPARISONS Parallel Lines Form inconsistent systems

COMPARISONS Coinciding Lines Have the same slope

COMPARISONS Coinciding Lines Have the same y-intercept

COMPARISONS Coinciding Lines Have infinite solutions

COMPARISONS Coinciding Lines Form a dependent system

A HELPFUL HINT For

A HELPFUL HINT For

Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.1 Let’s work Exercise #19 from Section 5.1

Section 8.1 Exercise #1 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.1 Exercise #1 Let’s work Exercise #19 from Section 5.1

Use the graphical method to solve the system. Let’s work Exercise #19 from Section 5.1

Use the graphical method to solve the system. x y 0 2 4 0 Let’s work Exercise #19 from Section 5.1

Use the graphical method to solve the system. x y 0 0 4 2 Let’s work Exercise #19 from Section 5.1

Section 8.1 Exercise #2 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.1 Exercise #2 Let’s work Exercise #19 from Section 5.1

Use the graphical method to solve the system. Let’s work Exercise #19 from Section 5.1

Use the graphical method to solve the system. x y 0 2 –1 0 Let’s work Exercise #19 from Section 5.1

Use the graphical method to solve the system. x y 0 4 –2 0 Inconsistent: No Solution. Let’s work Exercise #19 from Section 5.1

Section 8.2 Solving Systems of Equations by Substitution

OBJECTIVES Solve a system of equations in two variables. A

OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent. B

OBJECTIVES Solve an application. C

PROCEDURE: Solving a system of equations by the Substitution Method. 1. Solve one of the equations for x or y.

PROCEDURE: Solving a system of equations by the Substitution Method. Substitute the resulting expression into the other equation.

PROCEDURE: Solving a system of equations by the Substitution Method. 3. Solve the new equation for the variable.

PROCEDURE: Solving a system of equations by the Substitution Method. 4. Substitute the value of the variable and solve to get the value for the second variable.

PROCEDURE: Solving a system of equations by the Substitution Method. 5. Check the solution by substituting the numerical values of the variables in both equations.

Section 8.2 Exercise #3 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.2 Exercise #3 Let’s work Exercise #19 from Section 5.1

Use the method of substitution to solve the system (if possible). Let’s work Exercise #19 from Section 5.1 No Solution (inconsistent)

Section 8.2 Exercise #4 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.2 Exercise #4 Let’s work Exercise #19 from Section 5.1

Use the method of substitution to solve the system (if possible). Let’s work Exercise #19 from Section 5.1 Dependent (infinitely many solutions).

Section 8.3 Solving Systems of Equations by Elimination

OBJECTIVES Solve a system of equations in two variables. A

OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent. B

OBJECTIVES Solve an application. C

ELIMINATION METHOD One or both equations in a system of simultaneous equations can be multiplied (or divided) by any nonzero number to obtain an equivalent system.

ELIMINATION METHOD In the equivalent system, the coefficients of x (or y) are opposites, thus eliminating x or y when the equations are added.

Section 8.3 Exercise #5 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.3 Exercise #5 Let’s work Exercise #19 from Section 5.1

Solve the system (if possible). Multiply by –1 Multiply by 3 + Let’s work Exercise #19 from Section 5.1

Solve the system (if possible). (–1, –2) Let’s work Exercise #19 from Section 5.1

Section 8.3 Exercise #6 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.3 Exercise #6 Let’s work Exercise #19 from Section 5.1

Solve the system (if possible). Multiply by 2 + No solution (inconsistent) Let’s work Exercise #19 from Section 5.1

Section 8.4 Coin, General, Motion, and Investment Problems

OBJECTIVES Solve word problems involving coins. A Solve word problems of a general nature. B

OBJECTIVES Solve word problems using the distance formula: D = RT. C

OBJECTIVES Solve word problems involving the interest formula: I = Pr

Section 8.4 Exercise #8 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.4 Exercise #8 Let’s work Exercise #19 from Section 5.1

Eva has $2 in nickels and dimes. She has twice as many dimes as nickels. How many nickels and how many dimes does she have? Let’s work Exercise #19 from Section 5.1 Multiply by 100

She has 8 nickels and 16 dimes. Eva has $2 in nickels and dimes. She has twice as many dimes as nickels. How many nickels and how many dimes does she have? Divide by 25 Let’s work Exercise #19 from Section 5.1 She has 8 nickels and 16 dimes.

Section 8.4 Exercise #9 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.4 Exercise #9 Let’s work Exercise #19 from Section 5.1

The sum of two numbers is 140. Their difference is 90 The sum of two numbers is 140. Their difference is 90. What are the numbers? Let’s work Exercise #19 from Section 5.1 The numbers are 115 and 25.

Section 8.4 Exercise #10 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.4 Exercise #10 Let’s work Exercise #19 from Section 5.1

A plane flies 600 miles with a tailwind in 2 hours A plane flies 600 miles with a tailwind in 2 hours. It takes the same plane 3 hours to fly the 600 miles when flying against the wind. What is the plane’s speed in still air? Divide by 2 Let’s work Exercise #19 from Section 5.1

A plane flies 600 miles with a tailwind in 2 hours A plane flies 600 miles with a tailwind in 2 hours. It takes the same plane 3 hours to fly the 600 miles when flying against the wind. What is the plane’s speed in still air? Let’s work Exercise #19 from Section 5.1 Divide by 3

A plane flies 600 miles with a tailwind in 2 hours A plane flies 600 miles with a tailwind in 2 hours. It takes the same plane 3 hours to fly the 600 miles when flying against the wind. What is the plane’s speed in still air? Let’s work Exercise #19 from Section 5.1

The plane’s speed in still air is 250 mi/hr. Let’s work Exercise #19 from Section 5.1

Section 8.4 Exercise #11 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.4 Exercise #11 Let’s work Exercise #19 from Section 5.1

Herbert invests $10,000, part at 5% and part at 6% Herbert invests $10,000, part at 5% and part at 6%. How much money is invested at each rate if his annual interest is $568? Multiply by –5 Let’s work Exercise #19 from Section 5.1

Herbert invests $10,000, part at 5% and part at 6% Herbert invests $10,000, part at 5% and part at 6%. How much money is invested at each rate if his annual interest is $568? Multiply by 100 Let’s work Exercise #19 from Section 5.1

Herbert invests $10,000, part at 5% and part at 6% Herbert invests $10,000, part at 5% and part at 6%. How much money is invested at each rate if his annual interest is $568? Let’s work Exercise #19 from Section 5.1

$3,200 is invested at 5% and $6,800 is invested at 6%. Let’s work Exercise #19 from Section 5.1

Section 8.5 Systems of Linear Inequalities

OBJECTIVES Solve a system of linear inequalities by graphing. A

PROCEDURE: Solving a System of Inequalities. Graph each inequality on the same set of axes using the following steps.

PROCEDURE: Solving a System of Inequalities.

PROCEDURE: Solving a System of Inequalities. 2. Use any point (a,b) not on the line as a test point. Substitute the values of a and b for x and y.

PROCEDURE: Solving a System of Inequalities. If a true statement results, shade the side of the line containing the test point.

PROCEDURE: Solving a System of Inequalities. If a false statement results, shade the other side.

PROCEDURE: Solving a System of Inequalities. The solution set is the set of points that satisfies all the inequalities in the system.

Section 8.5 Exercise #12 Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.5 Exercise #12 Let’s work Exercise #19 from Section 5.1

Graph the solution set of the system. and Let’s work Exercise #19 from Section 5.1

Graph the solution set of the system. (0, –2) (4, 0) Let’s work Exercise #19 from Section 5.1

Graph the solution set of the system. (3, 0) (0, –2) (4, 0) Let’s work Exercise #19 from Section 5.1 (0, –6)

Graph the solution set of the system. and (3, 0) (0, –2) (4, 0) Let’s work Exercise #19 from Section 5.1 (0, –6)