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6.2 Using Substitution to Solve Systems Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 1 Example: Making a Substitution for y Solve the system y x3 2 x 3 y 14 Copyright © 2015, 2008 Pearson Education, Inc. Equation (1) Equation (2) Section 6.2, Slide 2 Solution From equation (1). We know that for any solution of the system, the value of y is equal to the value of x + 3. So, we substitute x + 3 for y in equation (2): 2 x 3 y 14 2 x 3 x 3 14 Copyright © 2015, 2008 Pearson Education, Inc. Equation (2) Section 6.2, Slide 3 Solution Note that, by making this substitution, we now have an equation in one variable. Next, we solve that equation for x: 2 x 3 x 3 14 2 x 3x 9 14 5x 9 14 5x 5 x 1 Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 4 Solution Thus the x-coordinate of the solution is 1. To find the y-coordinate, we substitute 1 for x in either of the original equations and solve for y: y x3 y 1 3 y4 Copyright © 2015, 2008 Pearson Education, Inc. Equation (1) Section 6.2, Slide 5 Solution So, the solution is (1, 4). We can check that (1, 4) satisfies both of the system’s equations: Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 6 Solution Or, we can verify that (1, 4) is the solution by graphing equations (1) and (2) and checking that (1, 4) is the intersection point of the two lines. To do so on a graphing calculator, we must first solve equation (2) for y: Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 7 Solution Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 8 Using Substitution to Solve a Linear System To use substitution to solve a system of two linear equations, 1. Isolate a variable on one side of either equation. 2. Substitute the expression for the variable found in step 1 into the other equation. 3. Solve the equation in one variable found in step 2. 4. Substitute the solution found in step 3 into one of the original equations, and solve for the other variable. Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 9 Example: Isolating a Variable and Then Using Substitution Solve the system 3x y 7 2x 5 y 1 Copyright © 2015, 2008 Pearson Education, Inc. Equation (1) Equation (2) Section 6.2, Slide 10 Solution We begin by solving for one of the variables in one of the equations. We can avoid fractions by choosing to solve equation (1) for y: 3x y 7 Equation (1) y 3x 7 Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 11 Solution Next, we substitute –3x – 7 for y in equation (2) and solve for x: 2x 5 y 1 Equation (2) 2 x 5 3x 7 1 2 x 15x 35 1 17 x 35 1 17 x 34 x 2 Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 12 Solution Finally, we substitute –2 for x in the equation y = –3x – 7 and solve for y: y 3 2 7 y 67 y 1 The solution is (–2, –1). We could then verify our work by checking that (–2, –1) satisfies both of the original equations. Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 13 Example: Applying Substitution to an Inconsistent System Consider the linear system y 3x 2 y 3x 4 Equation (1) Equation (2) The graphs of the equations are parallel lines (why?), so the system is inconsistent and the solution is the empty set. What happens when we solve this system by substitution? Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 14 Solution We substitute 3x + 2 for y in equation (2) and solve for x: y 3x 4 3x 2 3x 4 24 false Equation (2) We get the false statement 2 = 4. Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 15 Inconsistent System of Equations If the result of applying substitution to a system of equations is a false statement, then the system is inconsistent; that is, the solution set is the empty set. Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 16 Example: Applying Substitution to an Dependent System In an example in Section 6.1, we found that the system y 3x 5 Equation (1) 6 x 2 y 10 Equation (2) is dependent and that the solution set is the infinite set of solutions of the equation y = 3x – 5. What happens when we solve this system by substitution? Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 17 Solution We substitute 3x – 5 for y in equation (2) and solve for x: 6 x 2 y 10 6 x 2 3 x 5 10 6 x 6 x 10 10 10 10 true Equation (2) We get the true statement 10 = 10. Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 18 Dependent System of Equations If the result of applying substitution to a system of equations is a true statement that can be put into the form a = a, then the system is dependent; that is, the solution set is the set of ordered pairs represented by every point on the (same) line. Copyright © 2015, 2008 Pearson Education, Inc. Section 6.2, Slide 19