Two-Variable Linear System 7.2 Two-Variable Linear System Copyright © Cengage Learning. All rights reserved.

Objectives Use the method of elimination to solve systems of linear equations in two variables. Interpret graphically the numbers of solutions of systems of linear equations in two variables. Use systems of linear equations in two variables to model and solve real-life problems.

The Method of Elimination

The Method of Elimination We have studied two methods for solving a system of equations: substitution and graphing. Now, we will study the method of elimination. The key step in this method is to obtain, for one of the variables, coefficients that differ only in sign so that adding the equations eliminates the variable. Equation 1 Equation 2 Add equations.

The Method of Elimination Note that by adding the two equations, you eliminate the x-terms and obtain a single equation in y. Solving this equation for y produces y = 2, which you can then back-substitute into one of the original equations to solve for x.

The Method of Elimination

Example 2 – Solving a System of Equations by Elimination Solve the system of linear equations. Equation 1 Equation 2

The Method of Elimination In Example 2, the two systems of linear equations (the original system and the system obtained by multiplying by a constants) are called equivalent systems because they have precisely the same solution set.

The Method of Elimination The operations that can be performed on a system of linear equations to produce an equivalent system are (1) interchanging any two equations, (2) multiplying an equation by a nonzero constant, and (3) adding a multiple of one equation to any other equation in the system.

Graphical Interpretation of Solutions

Graphical Interpretation of Solutions It is possible for a general system of equations to have exactly one solution, two or more solutions, or no solution. If a system of linear equations has two different solutions, then it must have an infinite number of solutions.

Graphical Interpretation of Solutions A system of linear equations is consistent when it has at least one solution. A consistent system with exactly one solution is independent, whereas a consistent system with infinitely many solutions is dependent. A system is inconsistent when it has no solution.

Example 5 – Recognizing Graphs of Linear Systems Match each system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent.

Applications

Applications At this point, you may be asking the question “How can I tell whether I can solve an application problem using a system of linear equations?” The answer comes from the following considerations. 1. Does the problem involve more than one unknown quantity? 2. Are there two (or more) equations or conditions to be satisfied? If the answer to one or both of these questions is “yes,” then the appropriate model for the problem may be a system of linear equations.

Example 8 – An Application of a Linear System An airplane flying into a headwind travels the 2000-mile flying distance between Chicopee, Massachusetts, and Salt Lake City, Utah, in 4 hours and 24 minutes. On the return flight, the airplane travels this distance in 4 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant.

7.2 Example – Worked Solutions

Example 2 – Solving a System of Equations by Elimination Solve the system of linear equations. Solution: To obtain coefficients that differ only in sign, multiply Equation 2 by 4. Equation 1 Equation 2 Write Equation 1. Multiply Equation 2 by 4. Add equations. Solve for x.

Example 2 – Solution cont’d Solve for y by back-substituting into Equation 1. 2x – 4y = –7 –4y = –7 –4y = –6 The solution is Write Equation 1. Substitute for x. Simplify. Solve for y.

Example 2 – Solution cont’d Check this in the original system, as follows. Check 2x – 4y = –7 –1 – 6 = –7 5x + y = –1 Write Equation 1. Substitute for x and y. Solution checks in Equation 1. Write Equation 2. Substitute for x and y. Solutions checks in Equation 2.

Example 5 – Recognizing Graphs of Linear Systems Match each system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent.

Example 5 – Solution a. The graph of system (a) is a pair of parallel lines (ii). The lines have no point of intersection, so the system has no solution. The system is inconsistent. b. The graph of system (b) is a pair of intersecting lines (iii). The lines have one point of intersection, so the system has exactly one solution. The system is consistent. c. The graph of system (c) is a pair of lines that coincide (i). The lines have infinitely many points of intersection, so the system has infinitely many solutions. The system is consistent.

Example 8 – An Application of a Linear System An airplane flying into a headwind travels the 2000-mile flying distance between Chicopee, Massachusetts, and Salt Lake City, Utah, in 4 hours and 24 minutes. On the return flight, the airplane travels this distance in 4 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant. Solution: The two unknown quantities are the speeds of the wind and the plane.

Example 8 – Solution cont’d If r1 is the speed of the plane and r2 is the speed of the wind, then r1 – r2 = speed of the plane against the wind r1 + r2 = speed of the plane with the wind as shown in Figure 7.7. Figure 7.7

Example 8 – Solution cont’d Using the formula distance = (rate)(time) for these two speeds, you obtain the following equations. 2000 = (r1 – r2) 2000 = (r1 + r2)(4) These two equations simplify as follows. Equation 1 Equation 2

Example 8 – Solution cont’d To solve this system by elimination, multiply Equation 2 by 11. Write Equation 1. Multiply Equation 2 by 11. Add equations.

Example 8 – Solution So, miles per hour and miles per hour. cont’d Speed of plane Speed of wind