14.1 Systems of Linear Equations in Two Variables
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1 86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination to solve sstems of linear equations in two variables. Interpret graphicall the numbers of solutions of sstems of linear equations in two variables. Use sstems of linear equations in two variables to model and solve real-life problems. The Method of Substitution Up to this point in the tet, most problems have involved either a function of one variable or a single equation in two variables. However, man problems in science, business, and engineering involve two or more equations in two or more variables. To solve such problems, ou need to find solutions of a sstem of equations. Here is an eample of a sstem of two equations in two unknowns. 5 3 Equation A solution of this sstem is an ordered pair that satisfies each equation in the sstem. Finding the set of all solutions is called solving the sstem of equations. For instance, the ordered pair, 1 is a solution of this sstem. To check this, ou can substitute for and 1 for in each equation. Check (, 1) in and Equation : 5 1? ? Write. Substitute for and 1 for. Solution checks in. Write Equation. Substitute for and 1 for. 6 Solution checks in Equation. In this chapter, ou will stud four was to solve sstems of equations, beginning with the method of substitution. The guidelines for solving a sstem of equations b the method of substitution are summarized below. GUIDELINES FOR SOLVING A SYSTEM OF EQUATIONS BY THE METHOD OF SUBSTITUTION 1. Solve one of the equations for one variable in terms of the other.. Substitute the epression found in Step 1 into the other equation to obtain an equation in one variable. 3. Solve the equation obtained in Step.. Back-substitute the value obtained in Step 3 into the epression obtained in Step 1 to find the value of the other variable. 5. Check that the solution satisfies each of the original equations.
2 1.1 Sstems of Linear Equations in Two Variables 863 The term back-substitution implies that ou work backwards. First ou solve for one of the variables, and then ou substitute that value back into one of the equations in the sstem to find the value of the other variable. The back-substitution reduces the two-equation sstem to one equation in a single variable. EXAMPLE 1 Solving a Sstem of Equations b Substitution EXPLORATION Use a graphing utilit to graph 1 and in the same viewing window. Use the zoom and trace features to find the coordinates of the point of intersection. What is the relationship between the point of intersection and the solution found in Eample 1? Solve the sstem of equations. Solution Begin b solving for in. Solve for in. Net, substitute this epression for into Equation and solve the resulting single-variable equation for. Write Equation. Substitute for. Distributive Propert 6 Combine like terms. 3 Divide each side b. Finall, ou can solve for b back-substituting 3 into the equation, to obtain Write revised. 3 Substitute 3 for. 1. Solve for. The solution is the ordered pair 3, 1. You can check this solution as follows. Check Substitute 3, 1 into : Equation Write. 3 1? Substitute for and. Solution checks in. Substitute 3, 1 into Equation : 3 1? Write Equation. Substitute for and. Solution checks in Equation. Because 3, 1 satisfies both equations in the sstem, it is a solution of the sstem of equations. STUDY TIP Because man steps are required to solve a sstem of equations, it is ver eas to make errors in arithmetic. So, ou should alwas check our solution b substituting it into each equation in the original sstem.
3 86 Chapter 1 Sstems of Equations and Matrices The equations in Eample 1 are linear. The method of substitution can also be used to solve sstems in which one or both of the equations are nonlinear. Such a sstem ma have more than one solution. EXPLORATION Use a graphing utilit to graph the two equations in Eample in the same viewing window. How man solutions do ou think this sstem has? Repeat this eperiment for the equations in Eample 3. How man solutions does this sstem have? Eplain our reasoning. EXAMPLE Substitution: Two-Solution Case Solve the sstem of equations Solution Begin b solving for in Equation to obtain 1. Net, substitute this epression for into and solve for Equation Substitute 1 for in. Simplif. Write in general form. Factor. 3, Solve for. Back-substituting these values of to solve for the corresponding values of produces the solutions 3, 11 3 and, 3. Check these in the original sstem. The sstem of equations in Eample has two solutions. It is possible that a sstem has no solutions, as shown in Eample 3. EXAMPLE 3 Substitution: No Real-Solution Case Solve the sstem of equations. 3 Solution. Begin b solving for in to obtain Net, substitute this epression for into Equation and solve for ± ± 3 Equation Substitute for in Equation. Simplif. Quadratic Formula Simplif. Because the discriminant is negative, the equation 1 0 has no (real) solution. So, the original sstem has no (real) solution.
4 1.1 Sstems of Linear Equations in Two Variables 865 The Method of Elimination So far, ou have studied one method for solving a sstem of equations: substitution. Now ou will stud the method of elimination. The ke step in this method is to obtain, for one of the variables, coefficients that differ onl in sign so that adding the equations eliminates the variable Equation 3 6 Add equations. Note that b adding the two equations, ou eliminate the -terms and obtain a single equation in. Solving this equation for produces, which ou can then back-substitute into one of the original equations to solve for. EXAMPLE Solving a Sstem of Equations b Elimination Solve the sstem of linear equations Equation Solution Because the coefficients of differ onl in sign, ou can eliminate the -terms b adding the two equations Write. Write Equation Add equations. Solve for. B back-substituting into, ou can solve for Write. Substitute for. Simplif. Solve for. The solution is, 1. Check this in the original sstem, as follows. Check 3 1? 6 5 1? Substitute into. checks. Substitute into Equation. Equation checks. NOTE Although ou could use either the method of substitution or the method of elimination to solve the sstem in Eample, ou ma find that the method of elimination is more efficient.
5 866 Chapter 1 Sstems of Equations and Matrices EXAMPLE 5 Solving a Sstem of Equations b Elimination STUDY TIP To obtain coefficients (for one of the variables) that differ onl in sign, ou often need to multipl one or both of the equations b suitabl chosen constants. Solve the sstem of linear equations Equation Solution For this sstem, ou can obtain coefficients of the -terms that differ onl in sign b multipling Equation b. 7 7 Write Multipl Equation b. 11 Add equations. So, ou can see that 1. B back-substituting this value of into, ou can solve for Write. Substitute 1 for. Combine like terms. Solve for. The solution is 1, 3. Check this in both equations in the original sstem. Check 1 3? ? Substitute into. checks. Substitute into Equation. Equation checks. In Eample 5, the two sstems of linear equations and 7 0 are called equivalent sstems because the have precisel the same solution set. The operations that can be performed on a sstem of linear equations to produce an equivalent sstem are (1) interchanging an two equations, () multipling an equation b a nonzero constant, and (3) adding a multiple of one equation to an other equation in the sstem. GUIDELINES FOR SOLVING A SYSTEM OF EQUATIONS BY THE METHOD OF ELIMINATION 1. Obtain coefficients for or that differ onl in sign b multipling all terms of one or both equations b suitabl chosen constants.. Add the equations to eliminate one variable and solve the resulting equation. 3. Back-substitute the value obtained in Step into either of the original equations and solve for the other variable.. Check that the solution satisfies each of the original equations.
6 1.1 Sstems of Linear Equations in Two Variables 867 Graphical Interpretation of Solutions It is possible for a general sstem of equations to have eactl one solution, two or more solutions, or no solution. If a sstem of linear equations has two different solutions, it must have an infinite number of solutions. GRAPHICAL INTERPRETATIONS OF SOLUTIONS For a sstem of two linear equations in two variables, the number of solutions is one of the following. Number of Solutions Graphical Interpretation Slopes of Lines 1. Eactl one solution The two lines intersect at one point. The slopes of the two lines are not equal.. Infinitel man solutions The two lines coincide (are identical). The slopes of the two lines are equal. 3. No solution The two lines are parallel. The slopes of the two lines are equal. A sstem of linear equations is consistent if it has at least one solution. A consistent sstem with eactl one solution is independent, whereas a consistent sstem with infinitel man solutions is dependent. A sstem is inconsistent if it has no solution. EXAMPLE 6 Recognizing Graphs of Linear Sstems Match each sstem of linear equations with its graph. Describe the number of solutions and state whether the sstem is consistent or inconsistent a. b. c i. ii. iii. Solution a. The graph of sstem (a) is a pair of parallel lines (ii). The lines have no point of intersection, so the sstem has no solution. The sstem is inconsistent. b. The graph of sstem (b) is a pair of intersecting lines (iii). The lines have one point of intersection, so the sstem has eactl one solution. The sstem is consistent. c. The graph of sstem (c) is a pair of lines that coincide (i). The lines have infinitel man points of intersection, so the sstem has infinitel man solutions. The sstem is consistent. STUDY TIP A comparison of the slopes of two lines gives useful information about the number of solutions of the corresponding sstem of equations. To solve a sstem of equations graphicall, it helps to begin b writing the equations in slope-intercept form. Tr doing this for the sstems in Eample 6.
7 868 Chapter 1 Sstems of Equations and Matrices In Eamples 7 and 8, note how ou can use the method of elimination to determine that a sstem of linear equations has no solution or infinitel man solutions. EXAMPLE 7 No-Solution Case: Method of Elimination + = = 3 Figure 1.1 Solve the sstem of linear equations. 3 1 Equation Solution To obtain coefficients that differ onl in sign, ou can multipl b Multipl b. Write Equation. False statement Because there are no values of and for which 0 7, ou can conclude that the sstem is inconsistent and has no solution. The lines corresponding to the two equations in this sstem are shown in Figure 1.1. Note that the two lines are parallel and therefore have no point of intersection. In Eample 7, note that the occurrence of a false statement, such as 0 7, indicates that the sstem has no solution. In the net eample, note that the occurrence of a statement that is true for all values of the variables, such as 0 0, indicates that the sstem has infinitel man solutions. EXAMPLE 8 Man-Solution Case: Method of Elimination Solve the sstem of linear equations (1, 1) (, 3) = Equation Solution To obtain coefficients that differ onl in sign, ou can multipl b. 1 Multipl b. 0 0 Write Equation. Add equations. Because the two equations are equivalent (have the same solution set), ou can conclude that the sstem has infinitel man solutions. The solution set consists of all points, ling on the line 1 Figure 1. as shown in Figure 1.. Letting a, where a is an real number, ou can see that the solutions of the sstem are a, a 1. In Eample 8, choose some values of a to find solutions of the sstem: for eample, if a 1, the solution is 1, 1, and if a, the solution is, 3. Then check these solutions in the original sstem.
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