Chapter 4: Summary Definition/Procedure Example 4.1 Solving Systems by Graphing A system of linear equations consists of two or more linear equations with the same variables. A solution of a system of two equations in two variables is an ordered pair that is a solution of each equation in the system. (p. 245) Determine whether (4, 2) is a solution of the system x 2y 8 3x 4y 4 x 2y 8 3x 4y 4 4 2(2) 8 Substitute. 3(4) 4(2) 4 Substitute. 4 4 8 12 8 4 8 8 True 4 4 True Since (4, 2) is a solution of each equation in the system, yes, it is a solution of the system. To solve a system by graphing, graph each line on the same axes. a) If the lines intersect at a single point, then this point is the solution of the system. The system is consistent. b) If the lines are parallel, then the system has no solution. We write the solution set as . The system is inconsistent. c) If the graphs are the same line, then the system has an infi nite number of solutions. The equations are dependent. (p. 246) Solve by graphing. y 1 2 x 2 5x 3y 1 The solution of the system is (2, 3). The system is x consistent. y 5 (2, 3) y x 2 12 5 5 5 5x 3y 1 4.2 Solving Systems by the Substitution Method Steps for Solving a System by the Substitution Method Step 1: Solve one of the equations for one of the variables. If possible, solve for a variable that has a coeffi cient of 1 or 1. Step 2: Substitute the expression in Step 1 into the other equation. The equation you obtain should contain only one variable. Step 3: Solve the equation in Step 2. Step 4: Substitute the value found in Step 3 into either of the equations to obtain the value of the other variable. Step 5: Check the values in the original equations. (p. 256) Solve by the substitution method. 7x 3y 8 x 2y 11 Step 1: Solve for x in the second equation since its coeffi cient is 1. x 2y 11 Step 2: Substitute 2y 11 for the x in the fi rst equation. 7(2y 11) 3y 8 Step 3: Solve the equation above for y. 7(2y 11) 3y 8 14y 77 3y 8 Distribute. 17y 77 8 Combine like terms. 17y 85 Add 77 to each side. y 5 Divide by 17. Step 4: Substitute y 5 into the equation in Step 1 to fi nd x. x 2(5) 11 Substitute 5 for y. x 10 11 Multiply. x 1 Step 5: The solution is (1, 5). Verify this by substituting (1, 5) into each of the original equations. www.mhhe.com/messersmith CHAPTER 4 Summary 299
messersmith_power_introductory_algebra_1e_ch4_7_10
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