Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 11 Systems of Equations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Solving Systems of Linear Equations by Graphing

Martin-Gay, Developmental Mathematics, 2e 33 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. This section focuses on solving systems of linear equations containing two equations in two variables. A solution of a system of linear equations in two variables is an ordered pair of numbers that is a solution of both equations in the system.

Martin-Gay, Developmental Mathematics, 2e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Determine whether (–3, 1) is a solution of the system x – y = – 4 2x + 10y = 4 Replace x with –3 and y with 1 in both equations. First equation: –3 – 1 = – 4 True Second equation: 2(–3) + 10(1) = – = 4 True Since the point (–3, 1) produces a true statement in both equations, it is a solution of the system. Example

Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Determine whether (4, 2) is a solution of the system 2x – 5y = – 2 3x + 4y = 4 Replace x with 4 and y with 2 in both equations. First equation: 2(4) – 5(2) = 8 – 10 = – 2 True Second equation: 3(4) + 4(2) = = 20 ≠ 4 False Since the point (4, 2) produces a true statement in only one equation, it is NOT a solution. Example

Martin-Gay, Developmental Mathematics, 2e 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Since a solution of a system of equations is a solution common to both equations, it is also a point common to the graphs of both equations. To find the solution of a system of two linear equations, we graph the equations and see where the lines intersect. Solving Systems of Equations by Graphing

Martin-Gay, Developmental Mathematics, 2e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the system of equations by graphing. 2x – y = 6 x + 3y = 10 x y First, graph 2x – y = 6. (0, -6) (3, 0) (6, 6) Second, graph x + 3y = 10. (1, 3) (-2, 4) (-5, 5) The lines APPEAR to intersect at (4, 2). (4, 2) continued Example

Martin-Gay, Developmental Mathematics, 2e 88 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 the two equations. First equation: 2(4) – 2 = 8 – 2 = 6 True Second equation: 4 + 3(2) = = 10 True The point (4, 2) checks, so it is the solution of the system. continued

Martin-Gay, Developmental Mathematics, 2e 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint Neatly drawn graphs can help when “guessing” the solution of a system of linear equations by graphing.

Martin-Gay, Developmental Mathematics, 2e 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the system of equations by graphing. –x + 3y = 6 3x – 9y = 9 x y First, graph – x + 3y = 6. (-6, 0) (0, 2) (6, 4) Second, graph 3x – 9y = 9. (0, -1) (6, 1) (3, 0) The lines APPEAR to be parallel. continued Example

Martin-Gay, Developmental Mathematics, 2e 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Although the lines appear to be parallel, we need to check their slopes. –x + 3y = 6 First equation 3y = x + 6 Add x to both sides. y = x + 2 D ivide both sides by 3. 3x – 9y = 9 Second equation –9y = –3x + 9 Subtract 3x from both sides. y = x – 1 Divide both sides by –9. Both lines have a slope of, so they are parallel and do not intersect. Hence, there is no solution to the system. continued

Martin-Gay, Developmental Mathematics, 2e 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the system of equations by graphing. x = 3y – 1 2x – 6y = –2 x y First, graph x = 3y – 1. (-1, 0) (5, 2) (7, -2) Second, graph 2x – 6y = –2. (-4, -1) (2, 1) The lines APPEAR to be identical. continued Example

Martin-Gay, Developmental Mathematics, 2e 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Although the lines appear to be identical, we need to check that their slopes and y-intercepts are the same. x = 3y – 1 First equation 3y = x + 1 Add 1 to both sides. 2x – 6y = – 2 Second equation –6y = – 2x – 2 Subtract 2x from both sides. Any ordered pair that is a solution of one equation is a solution of the other. This means that the system has an infinite number of solutions. y = x + Divide both sides by 3. y = x + Divide both sides by -6. continued

Martin-Gay, Developmental Mathematics, 2e 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. There are three possible outcomes when graphing two linear equations in a plane. One point of intersection—one solution Parallel lines—no solution Coincident lines—infinite number of solutions If there is at least one solution, the system is considered to be consistent. If the system defines distinct lines, the equations are independent. Identifying Special Systems of Linear Equations

Martin-Gay, Developmental Mathematics, 2e 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Possible Solutions of Linear Equations Consistent The equations are dependent. If the lines lie on top of each other, then the system has infinitely many solutions. The solution set is the set of all points on the line. Inconsistent The equations are independent. If the lines are parallel, then the system of equations has no solution because the lines never intersect. Consistent The equations are independent. If the lines intersect, the system of equations has one solution given by the point of intersection. GraphType of SystemNumber of Solutions Two lines intersect at one point. Parallel lines Lines coincide (3, 5)

Martin-Gay, Developmental Mathematics, 2e 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Since there are only three possible outcomes with two lines in a plane, we can determine how many solutions of the system there will be without graphing the lines. Change both linear equations into slope-intercept form. We can then easily determine if the lines intersect, are parallel, or are the same line. Identifying Special Systems of Linear Equations

Martin-Gay, Developmental Mathematics, 2e 17 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Without graphing, determine the number of solutions of the system. 3x + y = 1 3x + 2y = 6 Write each equation in slope-intercept form. 3x + y = 1 First equation y = –3x + 1 Subtract 3x from both sides. 3x + 2y = 6 Second equation 2y = –3x + 6 Subtract 3x from both sides. The lines are intersecting lines (since they have different slopes), so the system is consistent and independent and has one solution. Divide both sides by 2. Example

Martin-Gay, Developmental Mathematics, 2e 18 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Without graphing, determine the number of solutions of the system. 3x + y = 0 2y = –6x Write each equation in slope-intercept form. 3x + y = 0First equation y = –3x Subtract 3x from both sides. 2y = –6xSecond equation y = –3x Divide both sides by 2. The two lines are identical, so the system is consistent and dependent, and there are infinitely many solutions. Example

Martin-Gay, Developmental Mathematics, 2e 19 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Without graphing, determine the number of solutions of the system. 2x + y = 0 y = –2x + 1 Write each equation in slope-intercept form. 2x + y = 0First equation y = –2x Subtract 2x from both sides. y = –2x + 1Second equation is already in slope-intercept form. The two lines are parallel lines. Therefore, the system is inconsistent and independent, and there are no solutions. Example