Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 4 Systems of Equations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 4.1 Solving Systems of Linear Equations in Two Variables

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Systems of Linear Equations A system of linear equations consists of two or more linear equations. The solution of a system of two equations in two variables is an ordered pair (x, y) that makes both equations true.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1a Determine whether the ordered pair (  3, 1) is a solution of the system. Solution First equationSecond equation Since (  3, 1) makes both equations true, it is a solution. The solution set is {(  3, 1)}.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1b Determine whether the ordered pair (4, 2) is a solution of the system. Solution First equationSecond equation Since (4, 2) does not make both equations true, it is not a solution of the system.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall We can estimate the solutions of a system by graphing each equation on the same coordinate system and estimating the coordinates of any point of intersection. Solving a System by Graphing

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2a Solve the system by graphing. If the system has just one solution, estimate the solution. Solution Graph each equation. The lines intersect at one point (4, 2). Check in each equation. (4, 2) continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Although the solution to the system of equations appears to be (4, 2), you still need to check the answer by substituting x = 4 and y = 2 into both equations. First EquationSecond Equation The point (4, 2) checks, so it is the solution of the system.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2b Solve the system by graphing. If the system has just one solution, estimate the solution. Solution Graph each equation. The lines appear to be parallel. continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall To be sure the lines are parallel, write each equation slope-intercept form. First EquationSecond Equation The graphs of these equations have the same slope, but different y-intercepts. The lines are parallel and the system has no solution and is said to be inconsistent.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2c Solve the system by graphing. If the system has just one solution, estimate the solution. Solution Graph each equation. The graph of the equation appears to be in the same line. continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall This means the equations have identical solutions. Any ordered pair solution of one equation satisfies the other equation. These equation are said to be dependent equations and there is an infinite number of solutions to the system.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall One solution: consistent system independent equations No solution: inconsistent system independent equations Infinite number of solutions: consistent system dependent equations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Graphing the equations of a system by hand is often a good method of finding approximate solutions of a system, but it is not a reliable method of finding exact solutions. We turn instead to two algebraic methods of solving systems. We use the first method, the substitution method, to solve the system. Solving a System by Substitution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 Use the substitution method to solve the system. Solution The first equation is solved for y, we can substitute 3x – 6 for y in the second equation. –4x + 2y = –8 –4x + 2(3x – 6) = –8 Substitute 3x – 6 for y. –4x + 6x – 12 = –8 Distribute 2x – 12 = –8 2x = 4 x = 2 continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The x-coordinate of the solution is 2. To find the y- coordinate, we replace x with 2 in the first equation. The ordered pair solution is (2, 0). Check to see that (2, 0) satisfies both equations of the system.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Solving a System of Two Equations Using the Substitution Method 1)Solve one of the equations for one of its variables. 2)Substitute the expression for the variable found in Step 1 into the other equation. 3)Find the value of one variable by solving the equation from Step 2. 4)Find the value of the other variable by substituting the value found in Step 3 into the equation from Step 1. 5)Check the ordered pair solution in both original equations.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The elimination method, or addition method is a second algebraic technique for solving systems of equations. We rely on a version of the addition property of equality, which states that “equals added to equals are equal.”

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 5 Use the elimination method to solve the system. Solution Eliminate x by adding the two equations. continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the elimination method to solve the system. The y-coordinate of the solution is 2. To find the corresponding x coordinate, we replace y with 2 in either original equation. Let’s use the second equation. The ordered pair solution is (2, 3). Check to see that (2, 3) satisfies both equations of the system. We leave that to you.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Solving a System of Two Linear Equations Using the Elimination Method 1)Rewrite each equation in standard form, Ax + By = C. 2)If necessary, multiply one or both equations by some nonzero number so that the coefficients of a variable are opposites of each other. 3)Add the equations. 4)Find the value of one variable by solving the equation from Step 3. 5)Find the value of the second variable by substituting the value found in Step 4 into either original equation. 6)Check the proposed ordered pair solution in both original equations.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 6 Use the elimination method to solve the system. Solution We can eliminate y if we multiply the first equation by 5 and the second equation by 3. continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the elimination method to solve the system. To find y, we let x =  1 in either equation of the system. The ordered pair solution is (  1,  1). We leave the check to you.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 7 Use the elimination method to solve the system. Solution We can eliminate y if we multiply the first equation by  2. continued Now we add the equations.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the elimination method to solve the system. The resulting equation 0 =  4, is false for all values of y or x. Thus, the system has no solution. This system is inconsistent, and the graphs of the equations are parallel lines.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 8 Use the elimination method to solve the system. Solution We can eliminate y if we multiply the first equation by 4. continued Now we add the equations.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the elimination method to solve the system. The resulting equation 0 = 0, is true for all possible values of y or x. The two equations are equivalent. They have the same solution set and there are an infinite number of solutions.