Transcription of 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.
2 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.
3 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 .
4 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.
5 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. 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.
6 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 .
7 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. 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.
8 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 . True TrueThe solution is by using the graphing 2y 43x y 5 Skill Practice13, 42.
9 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)
10 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. 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.
