Systems of Linear Equations

1 (A system of two linearequations representingmore than one line)2. NNCIEOSTINST(A system havingno solution) (An ordered pair that satisfies bothequations in a system of twoequations) (A system of Equations thathas one or more solutions) (A system of two linearequations that represents onlyone line)In this chapterwe solve Systems of Linear Equations in two and threevariables. Some new terms are introduced in the first section of this each word to find a key word from this chapter. As a hint, thereis a clue for each word. Complete the word scramble to familiarize yourselfwith the key ofLinear Systems of Linear Equations by Systems of Equations by Using theSubstitution Systems of Equations by Using the of Systems of Linear Equations in of Linear Equations in Three Variables Systems of Linear Equations by Using Matrices and Cramer s Rule177IA33miL2872X_ch03_177-254 09:22:2006 02:15 PM Page 177 CONFIRMING PAGES178 Chapter 3 Systems of Linear EquationsIA1.

2 Solutions to Systems of Linear EquationsA Linear equation in two variables has an infinite number of solutions that forma line in a rectangular coordinate system . Two or more Linear Equations form asystem of Linear Equations . For example:A solution to a system of Linear equationsis an ordered pair that is a solutionto eachindividual Linear Solutions to a system of Linear EquationsDetermine whether the ordered pairs are solutions to the the ordered pair into both Equations : True TrueBecause the ordered pair is a solution to both Equations , it is asolution to the systemof the ordered pair into both Equations : TrueFalseBecause the ordered pair is not a solution to the second equation,it is nota solution to the system of whether the ordered pairs are solutions to the ()b.()4, 10 2, 1y 2x 183x 2y 8 Skill Practice10, 62 3x y 2 3102 1 62 2 x y 6 102 1 62 610, 621 2, 42 3x y 2 31 22 1 42 2 x y 6 1 22 1 42 61 2, 4210, 621 2, 42 3x y 2 x y 6 Example 1 2x 4y 10 x 3y 5 Section Systems of Linear Equations by to Systems of and InconsistentSystems of Linear Systems of LinearEquations by GraphingSkill Practice smiL2872X_ch03_177-254 09:22:2006 02:15 PM Page 178 CONFIRMING PAGESS ection Systems of Linear Equations by Graphing179 IAA solution to a system of two Linear equationsmay be interpreted graphically as a point of intersec-tion between the two lines.

3 Notice that the lines inter-sect at (Figure 3-1).2. Dependent and Inconsistent Systems of Linear EquationsWhen two lines are drawn in a rectangular coordinate system , three geometricrelationships are lines may intersect at exactly one lines may intersect at no point. This occurs if the lines are lines may intersect at infinitely many pointsalong the line. This occurs if the Equations represent the same line (the lines are coinciding).If a system of Linear Equations has one or more solutions, the system is saidto be a consistent system . If a Linear equation has no solution, it is said to be aninconsistent two Equations represent the same line, then all points along the line aresolutions to the system of Equations . In such a case, the system is characterized asa dependent system . An independent systemis one in which the two equationsrepresent different 2, 4245 4 5 3123 2 1 3 4 5451 1 2yx32x y 6( 2, 4)3x y 2 Figure 3-1 Solutions to Systems of Linear Equations in Two VariablesOne unique solutionNo solutionInfinitely many solutionsOne point of intersectionParallel linesCoinciding linesSystem is is is is is is 09:22:2006 02:15 PM Page 179 CONFIRMING PAGESIA3.

4 Solving Systems of Linear Equations by GraphingSolving a system of Linear Equations by GraphingSolve the system by graphing both Linear Equations and finding the point(s)of :To graph each equation, write the equation in slope-intercept form First equation:Second equation:Slope:Slope:From their slope-intercept forms, we see that the lines have different slopes,indicating that the lines must intersect at exactly one point. Using the slopeand y-intercept we can graph the lines to find the point of intersection (Figure 3-2). 23 y 23x 2 3y3 2x3 63 3y 2x 6 2x 3y 6 32y 32x 12y mx 3y 6y 32x 12 Example 2 Figure 3-245 4 5 3123 2 1 3 4 5451 1 2yx32(3, 4)Point of intersection2x 3y 6y x 3212 Skill Practice 4 5 3123 1 3 4 5451 1 2yx32 2( 2, 1)The point appears to be the point of intersection. This can beconfirmed by substituting and into both Equations .

5 True TrueThe solution is by using the graphing 2y 43x y 5 Skill Practice13, 42. 2x 3y 6 2132 31 42 6 6 12 6y 32x 12 4 32132 12 4 92 12y 4x 313, 42180 Chapter 3 Systems of Linear EquationsmiL2872X_ch03_177-254 09:22:2006 02:15 PM Page 180 CONFIRMING PAGESIATIP:In Example 2, the lines could also have been graphed by using the x- and y-intercepts or by using a table of points. However, the advantage of writing theequations in slope-intercept form is that we can compare the slopes and y-intercepts of each If the slopes differ, the lines are different and nonparallel and must crossin exactly one If the slopes are the same and the y-intercepts are different, the lines areparallel and do not If the slopes are the same and the y-intercepts are the same, the twoequations represent the same 4 5 3123 2 1 3 4 5451 1 2yx32(2, 0)4x 86y 3x 6 Figure 3-3 Solving a system of Linear Equations by GraphingSolve the system by :The first equation can be written as This is an equation of avertical line.

6 To graph the second equation, write the equation in slope-intercept equation:Second equation:The graphs of the lines are shown in Figure 3-3. The point of intersection is(2, 0). This can be confirmed by substituting (2, 0) into both Equations . True TrueThe solution is (2, 0). the system by graphing. 3x y 4 4 4ySkill Practice6102 3122 66y 3x 64122 84x 8 y 12x 1 6y6 3x6 66 x 2 6y 3x 64x 8x 8 6y 3x 6 4x 8 Example 3 Skill Practice 4 5 3123 1 3 4 5451 1 2yx32 2(1, 1)Section Systems of Linear Equations by Graphing181miL2872X_ch03_177-254 09:22:2006 02:15 PM Page 181 CONFIRMING PAGESIAS olving a system of Equations by GraphingSolve the system by :To graph the line, write each equation in slope-intercept equation:Second equation: y 13x 1 y 13x 2 6y6 2x6 66 3y3 x3 63 3y x 6 6y 2x 6 x 3y 6 6y 2x 6 x 3y 6 Example 4 Skill Practice solution; inconsistent system 45 4 5 3123 1 3 4 5451 1 2yx32 2 Because the lines have the same slope but different y-intercepts, they areparallel (Figure 3-4).

7 Two parallel lines do not intersect, which implies that thesystem has no solution. The system is the system by a system of Linear Equations by GraphingSolve the system by :Write the first equation in slope-intercept form. The second equation is alreadyin slope-intercept equation:Second equation: y 14x 2 4y4 x4 84 4y x 8y 14x 2 x 4y 8 y 14x 2 x 4y 8 Example 5 x y 3 21y x2 0 Skill Practice182 Chapter 3 Systems of Linear EquationsFigure 3-445 4 5 3123 1 3 4 5451 1 2yx32 2 y x 113 y x 213miL2872X_ch03_177-254 09:22:2006 02:15 PM Page 182 CONFIRMING PAGESIAN otice that the slope-intercept forms of thetwo lines are identical. Therefore, the equa -tions represent the same line (Figure 3-5).The system is dependent, and the solution tothe system of Equations is the set of all pointson the not all the ordered pairs in thesolution set can be listed, we can write thesolution in set-builder notation.

8 Furthermore,the Equations and represent the same line. Therefore, the solutionset may be written as or the system by graphing. x 2y 2 y 12x 1 Skill Practice51x, y2 0 x 4y , y2 0 y 14x 26y 14x 2x 4y 8 Skill Practice Answers5.{}; infinitely many solutions; dependent systemy 12x 11x, y2 045 4 5 3123 1 3 4 5451 1 2yx32 2 The solution to a system of Equations can be found by using either a Tracefeature or an Intersectfeature on a graphing calculator to find the pointof intersection between two example, consider the systemFirst graph the Equations together on the same viewing window. Recall thatto enter the Equations into the calculator, the Equations must be writtenwith the y-variable isolated. That is, be sure to solve for inspection of the graph, it appears that the solution is TheTraceoption on the calculator may come close to but may not showthe exact solution (Figure 3-6).

9 However, an Intersectfeature on a graphingcalculator may provide the exact solution (Figure 3-7). See your user s man-ual for further 1, 421 1, 42. 5x y 1 y 5x 1 2x y 6 y 2x 6 5x y 1 2x y 6 Calculator ConnectionsFigure 3-545 4 5 3123 1 3 4 5451 1 2yx32 2 y x 214 Section Systems of Linear Equations by Graphing183miL2872X_ch03_177-254 09:22:2006 02:15 PM Page 183 CONFIRMING PAGESIAS tudy Skills you proceed further in Chapter 3, make your test corrections for the Chapter 2 test. See Exercise 1of Section for the key system of Linear equationsb. Solution to a system of Linear equationsc. Consistent systemd. Inconsistent systeme. Dependent systemf. Independent systemConcept 1: Solutions to Systems of Linear EquationsFor Exercises 3 8, determine which points are solutions to the given , 12b14, 02,1 2, 32, 2x 9y 1 4x 3y 4y 12x 2 x 3y 3 x y 6x 2y 4a3, 92b10, 102,14, 72,y 3x 7 y 34x 10 y 4x 32x 7y 30 y 12x 5 y 8x 5 Using TraceUsing IntersectFigure 3-6 Figure 3-7(2, 11)1 1, 12,1 1, 132,,,1 1, 42a32, 5b10, 30214, 22, 16, 02, 12, 4210, 12, 14, 12, 19, 22184 Chapter 3 Systems of Linear !

10 Practice Problems e-Professors Self-Tests Videos NetTutorSection ExercisesmiL2872X_ch03_177-254 09:22:2006 02:15 PM Page 184 CONFIRMING PAGESIAC oncept 2: Dependent and Inconsistent Systems of Linear EquationsFor Exercises 9 14, the graph of a system of Linear Equations is whether the system is consistent or whether the system is dependent or the number of solutions to the 3: Solving Systems of Linear Equations by GraphingFor Exercises 15 32, solve the Systems of Equations by y 5x 4 3x 4y 16 x 2y 1 y 2x 3 4x 3y 12 2x y 4 4x 6y 6 x 3y 6 4x 2y 0 y 23x 1 y 13x 2 y 2x 3 4x 2y 2 3y 2x 3 3x y 12x y 4 5x 3y 6 y x 345 4 5 3123 1 3 4 5451 1 2yx32 245 4 5 3123 1 3 4 5451 1 2yx32 245 4 5 3123 1 3 4 5451 1 2yx32 245 4 5 3123 1 3 4 5451 1 2yx32 245 4 5 3123 1 3 4 5451 2yx32 2 145 4 5 3123 1 3 4 5451 1 2yx32 245 4 5 3123 2 1 3 4 5451 1 2yx3245 4 5 3123 2 1 3 4 5451 1 2yx3245 4 5 3123 2 1 3 4 5451 1 2yx32 Section Systems of Linear Equations by Graphing185miL2872X_ch03_177-254 09:22:2006 02.