Solving Systems of Linear Equations

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1 Section 2.3 Solving Systems of Linear Equations TERMINOLOGY 2.3 Previously Used: Equivalent Equations Literal Equation Properties of Equations Substitution Principle Prerequisite Terms: Coordinate Axes Your definition Ordered Pair Your definition New Terms to Learn: System of Linear Equations (SLE) Your definition Formal definition Example READING ASSIGNMENT 2.3 Sections 4.1, 4.2 and 4.3 READING AND SELF-DISCOVERY QUESTIONS What is a system of equations 83

2 Chapter 2 Equations 2. What is a solution to a system of equations 3. How do you solve a system of equations in two variables graphically 4. What is the elimination method for solving a system of linear equations 5. What is the substitution method for solving a system of linear equations KEY CONCEPTS 2.3 Types of Solutions to Systems of Two Linear Equations in Two Variables Systems of two linear equations in two variables can have one solution, no solution, or an infinite number of solutions. The same types of solutions exist for any system of linear equations. Limitation: There are other types of systems of equations that are non-linear. 84

3 METHODOLOGY 2.3 GRAPHING THE SOLUTION TO A SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES This means of solving a system of linear equations can give insight into what it means to be a solution to such systems of equations. Limitation/Caution: This method may only provide approximate results, based on the precision of the graph. Solve by graphing: Example 1 Example 2 + 3y = 1 2x + 2y = 6 Solve by graphing: 4x 2y = 6 3x + y = 2 Steps Discussion 1 Graph both lines Graph the lines represented by each equation. Feel free to rewrite the equations in the slope-intercept form if that makes it easier to graph them. Note that the slope-intercept format is an equation where the variable y has been isolated. Example y x y 3 2 Example x

4 Chapter 2 Equations Steps Discussion 2 Identify the point of intersection Identify the ordered pair that represents the point where the two graphed lines intersect. 4 3 y (-2, 1) 2 Example x Example 2 (Identify it on your previous graph) 3 Present the solution Present the solution using the ordered pair in terms of the variables in the equations. ( 2, 1) or x = 2 and y = 1 4 Validate Validate your work algebraically by substituting the solution into each given equation and verifying that each is a true expression. Example 1 Let x = 2, y = 1 x + 3y = 1 2x + 2y = (1) = 1 2( 2) + 2(1) = 6 1 = = 6 6 = 6 Example 2 86

5 METHODOLOGY 2.3 SOLVING A SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES BY SUBSTITUTION The purpose of this methodology is to solve a system of two equations in two variables by using the Substitution Principle twice. Limitation/Caution: When you solve one of the equations for one variable in terms of the other, you should not substitute the result back into that same equation. Example 1 Example 2 Solve the system: Solve the system: + 2y x x + y = x + 5y 7 Steps Discussion 1 Select a variable and an equation Select a variable for the first substitution and an equation to use to determine an expression for the selected variable + 2y 3x Select x and the first equation (we choose this equation because x has no coefficient so we won't have to divide it away). 2 Solve one of the equations Solve the selected literal equation for the selected variable x + 2y x 2y 3 Substitute expression Substitute the expression for the variable solved for in Step 2 in the other equation. 3x 3( 3 2 y) 87

6 Chapter 2 Equations Steps Discussion 4 Solve for first variable Solve the resulting equation in one variable for that variable 9 6y 6y y = 22 y = 2 5 Substitute first variable Substitute the value obtained in Step 4 in the equation you created in Step 2 x 2y x 2( 2) 6 Solve for second variable Solve the equation in Step 5 for the other variable x = x = 1 7 Present solution Write the solution for each variable. (1, 2) or x = 1 and y = 2 8 Validate Validate your work by substituting the values for the variables in both original equations x + 2y 1 + 2( 2) = 3 1 4= 3 3 and 3x 3(1) 5( 2)

7 METHODOLOGY 2.3 SOLVING A SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES BY ELIMINATION The purpose of this methodology is to solve a system of two equations in two variables by using the Addition Property of Equations to create a third equation in one variable that can be solved. Limitation/Caution: A system of linear equations may have no solution. Example 1 Example 2 Solve the system: Solve the system: + 2y 2x + 3y = 2 3x 5x 2 Steps Discussion 1 Select the variable Select the variable to eliminate between the two equations + 2y 3x Eliminate y 2 Multiply the equations by the appropriate numbers Multiply the two equations by numbers such that the coefficients of the selected variable differ only by a sign ( x + y = ) ( x y = ) x + 1y = 15 6x 1y = 26 3 Add the two equations Add the two equations to eliminate the selected variable 11x = 11 89

8 Chapter 2 Equations Steps Discussion 4 Solve the equation Solve the equation from Step 3 for the variable x = 1 5 Substitute first variable Substitute the value of the variable you solved for in Step 4 in either of the original equations x + 2y (1) + 2y 6 Solve for second variable Solve the equation you obtained in Step 5 for the remaining variable 2y 1 2y = 4 y = 2 7 Present solution Write the solution for each variable. (1, 2) or x = 1 and y = 2 8 Validate Validate your work by substituting the values for the variables in both original equations x + 2y 1 + 2( 2) = 3 1 4= 3 3 and 3x 3(1) 5( 2)

9 CRITICAL THINKING QUESTIONS What does a solution to a system of linear equations in two variables represent 2. What is the central idea behind the substitution method 3. What is the central idea behind the elimination method 4. If a system of linear equations has no solutions, how are the slopes of the equations related 5. If a system of linear equations has multiple solutions, how are the graphs of the equations related 6. Should graphical solutions to a system of linear equations have the same solutions as algebraic solutions Explain your answer. 7. How do you validate the solution to a system of linear equations 91

10 Chapter 2 Equations DEMONSTRATE YOUR UNDERSTANDING 2.3 Solve each system of equations, using the indicated method. 1. 5x 3y = 2 5x 4y = 1 (elimination method) y = 3 2x + 6y = 6 (by graphing) 4 y x

11 3. 3x + 4y = 23 5x 4y = 15 (elimination method) 4. 3x 6x + 2y = 1 (substitution method) 93

12 Chapter 2 Equations IDENTIFY AND CORRECT THE ERRORS 2.3 In the second column, identify the error you find in each of the following worked solutions and describe the error made. Solve the problem correctly in the third column. 1. Solve by elimination: Problem Describe Error Correct Process 2x 6x + y = 6 Worked Solution (What is wrong here) 2x 6x + y = 6 4x = 4 x = 1 The solution is x = Solve by elimination: 2x + 4y = 12 2x + 3y = 2 Worked Solution (What is wrong here) 2x + 4y = 12 2x + 3y = 2 7x = 14 x = 2 2(2) + 4y = y = 12 4y = 16 y = 4 The solution is (2, 4) or x = 2 and y = 4. 94

13 Problem Describe Error Correct Process 3. Solve by substitution: 2x 6x + y = 6 Worked Solution (What is wrong here) 2x 6x + y = 6 2x y = 2 + 2x 2 x ( x) = 2 2x + 2 2x = 2 = The solution set is all real numbers. 4. Solve by substitution: 5x = 6 Worked Solution (What is wrong here) 5x = 6 ( x y ) 5 = 2 5x = 6 5x + 5y = 1 5x = 6 = 16 The solution set is all real numbers. 95

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