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## Section 8.1

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**Section 8.1**Solving Systems of Linear Equations by Graphing**Solve a system of two equations in two variables by**graphing. A OBJECTIVES**B**OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent.**C**OBJECTIVES Solve an application.**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 samey-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**Chapter 8**Solving Systems of Linear Equations and Inequalities Section 8.1Exercise #1**x y0 24 0**Use the graphical method to solve the system.**x y0 04 2**Use the graphical method to solve the system.**Chapter 8**Solving Systems of Linear Equations and Inequalities Section 8.1Exercise #2**x y0 2–1 0**Use the graphical method to solve the system.**x y0 4–2 0**Use the graphical method to solve the system. Inconsistent:No Solution.**Section 8.2**Solving Systems of Equations by Substitution**Solve a system of equations in two variables.**A OBJECTIVES**B**OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent.**C**OBJECTIVES Solve an application.**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.**Chapter 8**Solving Systems of Linear Equations and Inequalities Section 8.2Exercise #3**Use the method of substitution to solve the system (if**possible). No Solution (inconsistent)**Chapter 8**Solving Systems of Linear Equations and Inequalities Section 8.2Exercise #4**Use the method of substitution to solve the system (if**possible). Dependent (infinitely many solutions).**Section 8.3**Solving Systems of Equations by Elimination**Solve a system of equations in two variables.**A OBJECTIVES**B**OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent.**C**OBJECTIVES Solve an application.**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 eliminatingx or y when the equations are added.**Chapter 8**Solving Systems of Linear Equations and Inequalities Section 8.3Exercise #5