ࡱ>  Oq LYbjbjt+t+ L@AAP]ttttxtt#,P  :$OChu  P r tt  4/̷0tt|rWorksheet 54 (10.1) Chapter 10 Systems of Equations 10.1 Systems of Two Linear Equations in Two Variables Summary 1: Systems of Equations A system of two linear equations in two variables is two equations considered together. To solve a system is to find all the ordered pairs that satisfy both equations. When solving a system three situations can occur: 1. The lines intersect. There is one solution, the point where they intersect. The system is called a consistent system. 2. The lines are parallel. There is no solution, the lines do not intersect. The system is called an inconsistent system. 3. The lines are the same. There are infinitely many solutions, the lines coincide. The system is called dependent. Systems of equations can be solved by various methods which will be discussed in this chapter. Warm-up 1. a) Use the graphing approach to solve the system. Tell whether the system is consistent, inconsistent or dependent. EMBED Equation.3 Graph the two lines using any method discussed in Chap. 7. y x The solution to the system is the point of intersection: ( , ). The system is called . Worksheet 54 (10.1) Problem 1. Use the graphing approach to solve the system. Tell whether the system is consistent, inconsistent or dependent.EMBED Equation.3 y x Summary 2: The Substitution Method To solve a system of equations by the substitution method: 1. Solve one of the equations for one of the variables.(Choose a variable with a coefficient of 1 or -1 if possible.) 2. Substitute this expression into the other equation to produce an equation with only one variable. 3. Solve the equation in Step 2 for the remaining variable. 4. Substitute this solution into the expression obtained in Step 1. 5. Solve for the second variable. 6. Write your solution set as an ordered pair, and check in each equation. Warm-up 2. a) Solve the system by the substitution method: EMBED Equation.3 y = 2 - 3x 1) Solve the 2nd equation for y. 2x - 3( ) = 5 2) Substitute 2 - 3x into the other equation in place of y. 2x - + = 5 3) Solve for x. 11x = x = y = 2 - 3( ) 4) Substitute this value into step 1. y = 5) Solve for y. {( , )} 6) Write the solution set. Check: 2( ) - 3( ) = 5 ? 7) Check the solution in both equations. 3( ) + ( ) = 2 ? Worksheet 54 (10.1) Problem 2. Solve the system by the substitution method: EMBED Equation.3 Warm-up 3. a) Solve by setting up a system of equations: Brian invested $10,000, part of it at 6% interest and the remainder at 4%. His yearly interest income from the two investments was $540. How much did he invest at each rate? It is sometimes helpful to set up a chart.  rate  amount = interest investment 1  6% = 0.06 x  0.06x investment 2 4% = 0.04 y  total  -------- 10,000  Fill in the last two entries in the chart. Then use the last two columns to write your system of equations. EMBED Equation.3 x = Solve 1st equation for x. 0.06( ) + 0.04y = 540 Substitute this expression into other equation in place of x. - + 0.04y = 540 Solve for y. -0.02y = - y = x = 10,000 - Sub. this value for y into 1st step. x = Solve for x Brian invested at 6% and at 4%. Worksheet 54 (10.1) Problem - Solve by setting up a system of equations: 3. Two angles are complementary and the measure of one of them is 30 more than the measure of the other. Find the measure of each angle. Worksheet 55 (10.2) 10.2 Elimination-by-Addition Method Summary 1: Elimination-by-Addition Method In the elimination-by-addition method we use the following operations to produce an equivalent system which can be easily solved. 1. Any two equations can be interchanged. 2. Both sides of an equation can be multiplied by any nonzero real number. 3. Any equation can be replaced by the sum of that equation and a nonzero multiple of another equation. Steps for elimination-by-addition method: 1. Write each equation in standard form if needed. 2. If necessary, multiply one or both equations by some constant which will make the x or y coefficients opposites. 3. Add the equations from step 2 together eliminating one of the variables. 4. Solve for the remaining variable. 5. Substitute this solution into either of the original equations. 6. Solve for the second variable. 7. Write the solution set and check. Warm-up 1. a) Solve the system by the elimination-by-addition method: EMBED Equation.3 The equations are in standard form. 3x = If we add the 2 equations the y's will cancel. x = Solve for x. + 3y = 7 Substitute x = 4 into either equation. 3y = Solve for y. y = {( , )} Write the solution set and check. check: 4 + 3(1) = 7 ? 2(4) - 3(1) = 5 ? Worksheet 55 (10.2) b) Solve the system by the elimination-by-addition method: EMBED Equation.3 Equations are in standard form. 9x - 6y = -63 Multiply the first equation by 3 and the 4x + 6y = -2 second equation by 2 so the coefficients of the y terms become -6 and 6. 13x = Add the two new equations together to eliminate the y terms. x = Solve for x. 3( ) - 2y = -21 Substitute this solution for x into either original equation. - 2y = -21 Solve for y. -2y = y = { ( , ) } Write the solution set and check. check: 3(-5) - 2(3) = -21 ? 2(-5) + 3(3) = -1 ? Summary 2: Which Method to Use? 1. If one equation is already solved for one of the variables, substitution would probably be the easiest method. 2. If solving for either variable in either equation would produce fractions to substitute, use the addition method. 3. Always clear fractions in both equations before deciding which method to use. Problems - Solve by either method: 1. EMBED Equation.3 Worksheet 55 (10.2) 2. EMBED Equation.3 Warm-up 2. a) Solve the system: EMBED Equation.3 4x - 2( ) = 12 Since the first equation is already solved for y, substitute 2x + 7 into the second equation in place of y. 4x - - = 12 Solve for x = 12 This is a false statement, which implies The solution set is the system has no solution and is an inconsistent system. Problem 3. Solve the system: EMBED Equation.3 FNote: In Problem 3 the solution set can be expressed {(x,y)2x + y = 9}. Obtaining an equation which is always true indicates that the two lines are the same and any point that satisfies the equations is a solution. The system is dependent. Worksheet 56 (10.3) 10.3 Systems of Three Linear Equations in Three Variables Summary 1: Linear Equations in Three Variables Linear equations in 3 variables such as x + y + z = 5 have solutions which are ordered triples, (x,y,z). The graph of a linear equation in 3 variables is a plane. To graph you would need a 3-dimensional coordinate system. To solve a system of 3 linear equations in 3 variables, means to find all the ordered triples that satisfy all three equations. Similar to a system of 2 equations: 1) There can be one solution. (All 3 planes intersect at one point.) 2) There can be many solutions. (The 3 planes intersect in more than one point.) 3) There can be no solution. (All 3 planes never intersect.) We will solve systems with 3 variables and 3 equations by the elimination- by-addition method. Warm-up 1. a) Solve the system by the elimination-by-addition method: EMBED Equation.3 -6y - 2z = 16 Multiply equation 2 by -2. y + 2z = -1 Add the equation obtained in the -5y = previous step to equation 3. y = Solve for y. 3( ) + z = -8 Substitute this solution into either equation containing only y and z. + z = -8 Solve for z. z = x + 2( ) - ( ) = -5 Substitute y and z into the first equation. x - - = -5 Solve for x. x - = -5 x = { ( , , ) } Write the solution set as an ordered triple. Checking this solution in each equation will be left to you. Worksheet 56 (10.3) b) Solve the system by the elimination-by-addition method: EMBED Equation.3 6x - 2y + 4z = -4 Multiply equation 3 by 2 and add to x + 2y - 5z = -29 equation 2, eliminating the y terms. 7x - z = -9x + 3y - 6z = 6 Multiply equation 3 by -3 and add to 2x - 3y + z = -3 equation 1 eliminating the y terms. -7x = 7x - z = -33 Add the equations resulting from the first -7x - 5z = 3 two steps, eliminating the x terms. -6z = Solve for z. z = 7x - ( ) = -33 Substitute this value for z into either equation containing only x and z. 7x = Solve for x. x = 2( ) - 3y + ( ) = -3 Substitute the solutions for x and z into any of the original equations. - 3y + 5 = -3 Solve for y. -3y = y = { ( , , ) } Write the solution set as an ordered triple. check: 2(-4) - 3(0) + 5 = -3 ? (-4) + 2(0) -5(5) = -29 ? 3(-4) - 0 + 2(5) = -2 ? Worksheet 56 (10.3) Problems - Solve the systems: 1. EMBED Equation.3 2. EMBED Equation.3 Worksheet 57 (10.4) 10.4 Matrix Approach to Solving Systems Summary 1: Matrices A matrix is an array of numbers arranged in horizontal rows and vertical columns. If a matrix has 2 rows and 3 columns it is called a 2 x 3 (two-by-three) matrix. A matrix can have any number of rows or columns. In general a matrix with m rows and n columns is called a matrix of dimension m x n. Every system of linear equations has associated with it an augmented matrix consisting of the coefficients and constant terms of the system. Summary 2: Gaussian Elimination A system of equations can be solved using the augmented matrix and the following Elementary Row Operations: 1. Any two rows of an augmented matrix can be interchanged. 2. Any row can be multiplied by a nonzero constant. 3. Any row of the augmented matrix can be replaced by adding a nonzero multiple of another row to that row. This method is called Gaussian Elimination. Warm-up 1. Solve the systems using the augmented matrix of the system: a) EMBED Equation.3 EMBED Equation.3 Write the augmented matrix. EMBED Equation.3 Interchange row 1 and row 2. EMBED Equation.3 Multiply row 1 by -3 and add to row 2 to produce a new row 2. Worksheet 57 (10.4) EMBED Equation.3 The matrix can be rewritten as a new system. 7y = 21 Solve the last equation for y. y = x - 2( ) = -7 Substitute the value for y into the first equation of the new system. x - = -7 Solve for x. x = {( , )} Write the solution set. b) EMBED Equation.3 EMBED Equation.3 Write the augmented matrix. EMBED Equation.3 Interchange row 3 and row 1. EMBED Equation.3 Multiply row 1 by -4 and add to row 2. EMBED Equation.3 Interchange row 2 and row 3. EMBED Equation.3 Multiply row 2 by -5 and add to row 3. EMBED Equation.3 Rewrite as a new system. FNote: This matrix represents a system that is in triangular form. Worksheet 57 (10.4) 19z = 19 Solve the 3rd equation for z. z = -y - ( ) = 1 Substitute this value for z into the 2nd equation. -y = Solve for y. y = x + ( ) - 3( ) = -7 Substitute z and y into the 1st equation. x - 2 - = -7 Solve for x. x - = -7 x = {( , , )} Write the solution set. Problems - Solve using the augmented matrix of the system. 1. EMBED Equation.3 Worksheet 55 (10.4) 2. EMBED Equation.3 Worksheet 58 (10.5) 10.5 Determinants Summary 1: Determinants A square matrix has the same number of rows as columns. For example, a 2 x 2 matrix or a 3 x 3 matrix. The determinant of a square matrix is a real number. For the square matrix EMBED Equation.3 the determinant is defined by: EMBED Equation.3 To evaluate the determinant, multiply the element in row 1, column 1 (a1) times the element in row 2, column 2 (b2). Subtract from this quantity the product of the element in row 2, column 1 (a2) and the element in row 1, column 2 (b1): EMBED Equation.3 The vertical bars used above are the algebraic symbols indicating to evaluate the determinant. Warm-up 1. a) Find the determinant of the matrix:EMBED Equation.3 determinant = EMBED Equation.3 = - = b) Evaluate: EMBED Equation.3 = - = Problems - Evaluate each determinant: 1. EMBED Equation.3 2. EMBED Equation.3 Worksheet 58 (10.5) Summary 2: Cramer's Rule Cramer's Rule is a method of using determinants to solve a system of 2 linear equations in 2 variables. Given the system: EMBED Equation.3, to use Cramer's Rule we need to calculate three determinants. D = determinant of the coefficients of x and y = EMBED Equation.3 Dx = determinant formed by replacing the x coefficients with the constants =EMBED Equation.3 Dy = determinant formed by replacing the y coefficients with the constants = EMBED Equation.3 ; then EMBED Equation.3 Warm-up 2. Use Cramer's Rule to solve the system: a) EMBED Equation.3 EMBED Equation.3 = + = EMBED Equation.3 = - = Worksheet 58 (10.5) EMBED Equation.3 = + = EMBED Equation.3 Solution Set = { ( , ) } b) EMBED Equation.3 EMBED Equation.3 = + = EMBED Equation.3 = - = EMBED Equation.3 = - = EMBED Equation.3 Solution Set = { ( , )} Problems - Use Cramer's Rule to solve the systems: 3. EMBED Equation.3 Worksheet 58 (10.5) 4. EMBED Equation.3 Worksheet 59 (10.6) 10.6 3 x 3 Determinants and Systems of Three Linear Equations Summary 1: The determinant of a 3 x 3 matrix is defined by: EMBED Equation.3 An easier way to calculate the determinant of a 3 x 3 matrix is a method called expansion of a determinant by minors. A minor of an element in a determinant is the determinant that remains after deleting the row and column in which the element appears. Warm-up 1. Use expansion by minors to evaluate the determinant: a)EMBED Equation.3 We will expand by minors on the first column. 1) Multiply the number in row 1, column 1 (2) times the determinant of the four numbers remaining if you cross out the row and column containing the 2. EMBED Equation.3 2) From this subtract the product of the number in row 2, column 1 (5) and the determinant of the four numbers remaining if you cross out the row and column containing the 5: EMBED Equation.3 3) To this add the product of the number in row 3, column 1 (1) and the determinant of the four numbers remaining if you cross out the row and column containing the 1. EMBED Equation.3 Worksheet 59 (10.6) EMBED Equation.3 =EMBED Equation.3EMBED Equation.3EMBED Equation.3 step 1 step 2 step 3 = 2(-14 - (-24)) - 5(2 - 18) + 1( - ) = 2( ) - 5( ) + 1( ) = + + = Problem - Use expansion by minors to evaluate the determinant: 1. EMBED Equation.3 Summary 2: Cramer's Rule for 3 x 3 Systems Given the system: EMBED Equation.3 with EMBED Equation.3 EMBED Equation.3 then EMBED Equation.3 Worksheet 59 (10.6) Warm-up 2. Use Cramer's Rule to find the solutions of the system: a) EMBED Equation.3 We will find the determinant D, Dx, Dy and Dz by expanding by minors on the first column. EMBED Equation.3 = 1(-15 - 8) - 2(5 - ( )) + (-3)( - ) = 1( ) - 2( ) + (-3)( ) = - - = EMBED Equation.3 = -2(-15 - 8) - 17(5 - ( )) - 7(4 - ) = -2( ) - 17( ) - 7( ) = - - = EMBED Equation.3 = 1(85 - ( )) - 2(-10 - ) - 3(-8 - ( )) = 1( ) - 2( ) - 3( ) = + - = EMBED Equation.3 = 1(21 - ) - 2( - ( )) - 3( - ) = 1( ) - 2( ) - 3( ) = + - = EMBED Equation.3 Solution Set = {( , , )} Worksheet 59 (10.6) Problem - Use Cramer's Rule to find the solution of the system: 2. EMBED Equation.3 Worksheet 60 (10.7) 10.7 Systems Involving Nonlinear Equations and Systems of Inequalities Summary 1: Systems of Nonlinear Equations Most nonlinear systems involve two quadratic equations or a quadratic equation and a linear equation. The graphs of the quadratic equations will be one of the four conic sections. Systems of this type are solved by the substitution method or the elimination method. Warm-up 1. a) Graph the following system to approximate the solutions, then solve by substitution or elimination method. EMBED Equation.3 y x The solutions should be a point in Quadrant I and a point in Quadrant III. By inspection, points of intersection could be (-1, -3) and (3, 5). Solve the system by substitution. EMBED Equation.3 = x2 - 4 Substitute 2x - 1 into equation 1 in place of y. 0 = x2 - - Set equal to zero. 0 = (x - 3)( ) Factor. x - 3 = 0 or = 0 Solve for x. x = or x = y = ( )2 - 4 or y = ( )2 - 4 Substitute each of these 2 values for x into either equation. y = y = Solve for the corresponding y's. Solution Set: {( , ), ( , )} Worksheet 60 (10.7) Problems - Graph the system to approximate the solutions then solve the system. 1. EMBED Equation.3 y x Summary 2: Systems of Linear Inequalities To find the solution set for a system of linear inequalities, graph each inequality on the same coordinate system. The intersection of the two graphs is the solution set for the system. 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CJOJQJUVhmHnH j5U5>*565>*CJH*8SV^V~VVCWDWEWFWGWHWIWLWWWWWWW$p&`t $d%d&d'd/.+D"$$@ $&`$d%d&d'd.&`j$d%d&d'd/. 5$WWX#XGXQXyXXXXXXXXXXXXXXY(Y)Y/Y6Y7Y8Y9Y$$ $$$pY$Y%Y&Y'Y)Y.Y2Y3Y4Y;Y~SƳy|'f b z Zfi̽jd1DdL <   C A? 2M"2dPO[ `!M"2dPO[8px]1K@߻K6)mQq \% 8`;  *-jtv ~~'AjA0޻KĔ{˽ #0Xz2b  PFj$# #o9[1K !%AYz* ;;kWD*YG6T_ `TMDsnU9++!KsP<Ou~ ?t~+W.TOֹuC9K{177tnF#geh81rJ;TíXhUS_Ew<]a [l[Ln R6yꜱdžAУ5G?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~PRoot Entry F0˷08̷0 Data WordDocumentL@ObjectPool˷08̷0_1060589559F˷0˷0Ole CompObjfObjInfo  #&'(*+.123567:=>?ABEHIJLMPSTUWX[^_`bcfijkmnqtuvxy| FMicrosoft Equation 3.0 DS Equation Equation.39q  2x -  y = 5 3x + 2y = 4 ()Ole10Native Equation Native _1060589558 FǪ˷0Ǫ˷0Ole  ߄@yII  2x -  y = 5 3x + 2y = 4 () FMicrosoft Equation 3.0 DS Equation Equation.39q  x + y = 5  x CompObj  fObjInfoOle10Native Equation Native - y = 3()t@yII  x + y = 5  x - y = 3() FMicrosoft Equation 3.0 DS Eq_1060589557FǪ˷0Ǫ˷0Ole CompObjfObjInfouation Equation.39q  2x - 3y = 5  3x +  y = 2 ()ߌ@yII  2x - 3y = 5  3x +  Ole10NativeEquation Native _1060589556" FǪ˷0Ǫ˷0Ole !y = 2 () FMicrosoft Equation 3.0 DS Equation Equation.39q  x - 2y = -12  2x + 9y = 2 ()CompObj"fObjInfo$Ole10Native%Equation Native )ߌ@yII  x - 2y = -12  2x + 9y = 2 () FMicrosoft Equation 3.0 DS Equation Equation.39q_1060589555FǪ˷0Ǫ˷0Ole ,CompObj -fObjInfo/Ole10Native!0Equation Native 4_1060589554($FǪ˷0Ǫ˷0Ole 8  x +     y = 10,000  0.06 x + 0.04 y = 540()@yII  x +     y = 10,000  0.06 x + 0.04 y = 540() FMicrosoft Equation 3.0 DS Equation Equation.39q x + 3y = 72x - 3y = 5()CompObj#&9fObjInfo;Ole10Native%'<Equation Native @x@yII x + 3y = 72x - 3y = 5() FMicrosoft Equation 3.0 DS Equation Equation.39q  3x - 2y = -21_1060589553*FǪ˷0Ǫ˷0Ole CCompObj),DfObjInfoFOle10Native+-GEquation Native K_1060589552F0FǪ˷0Ǫ˷0Ole N2x + 3y = -1()ߌ@yII  3x - 2y = -212x + 3y = -1() FMicrosoft Equation 3.0 DS EqCompObj/2OfObjInfoQOle10Native13REquation Native Vuation Equation.39q  5x -  y  = 13 3x + 2y = 0 ()ߐ@yII  5x -  y  = 13 3x + 2y = 0 () FMicrosoft Equation 3.0 DS Equation Equation.39q  4x - 3y = 19  5x + 4y = -15 ()_10605895516Fh˷0h˷0Ole YCompObj58ZfObjInfo\Ole10Native79]Equation Native a_1060589550@4<Fh˷0h˷0Ole dߘ@yII  4x - 3y = 19  5x + 4y = -15 () FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj;>efObjInfogOle10Native=?hEquation Native l  y = 2x + 7 4x - 2y = 12()߀@yII  y = 2x + 7 4x - 2y = 12()_1060589549BFh˷0h˷0Ole oCompObjADpfObjInfor FMicrosoft Equation 3.0 DS Equation Equation.39q 2x +  y = 9 4x + 2y = 18 ()߄@yII 2x + Ole10NativeCEsEquation Native w_1060589548R:HFh˷0h˷0Ole z y = 9 4x + 2y = 18 () FMicrosoft Equation 3.0 DS Equation Equation.39q  x + 2y -  z = -5  3y +  z = -8 y CompObjGJ{fObjInfo}Ole10NativeIK~Equation Native + 2z = -1()@yII  x + 2y -  z = -5  3y +  z = -8 y + 2z = -1() FMicrosoft Equation 3.0 DS Eq_1060589547NFh˷0h˷0Ole CompObjMPfObjInfouation Equation.39q  2x - 3y +  z = -3 x + 2y - 5z = -293x -  y + 2z = -2()@yII  2x - 3y +  z = -3 xOle10NativeOQEquation Native _1060589546XLTF`˷0`˷0Ole  + 2y - 5z = -293x -  y + 2z = -2() FMicrosoft Equation 3.0 DS Equation Equation.39q  x - y - 3z = CompObjSVfObjInfoOle10NativeUWEquation Native 4 2y + 4z = 0 y - 3z = -10 ()@yII  x - y - 3z = 4 2y + 4z = 0 y - 3z = -10 ()_1060589545ZF`˷0`˷0Ole CompObjY\fObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q  2x + 5y - 3z = 16  x  -  y -  z = -5  3x + 2y +  z = -20 ()Ole10Native[]Equation Native (_1060589544.`F`˷0`˷0Ole  @yII  2x + 5y - 3z = 16  x  -  y -  z = -5  3x + 2y +  z = -20 ()CompObj_bfObjInfoOle10NativeacEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q  3x +  y = 0  x - 2y = -7 ()߈@yII  3x +  y = 0  x - 2y = -7 () FMicrosoft Equation 3.0 DS Equation Equation.39q  3   1    0 1  -2    -7 []_1060589543fF`˷0`˷0Ole CompObjehfObjInfoOle10NativegiEquation Native _1060589542pdlF`˷0`˷0Ole ߈@yII  3   1    0 1  -2    -7 [] FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjknfObjInfoOle10NativemoEquation Native   1   -2   -7 3    1    0[]߈@yII  1   -2   -7 3    1    0[]_1060589541rF`˷0`˷0Ole CompObjqtfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q  1  -2  -7 0   7   21[]Ole10NativesuEquation Native _1060589540jxF1˷01˷0Ole |@yII  1  -2  -7 0   7   21[] FMicrosoft Equation 3.0 DS Equation Equation.39q  x - 2y = -7  CompObjwzfObjInfoOle10Nativey{Equation Native 7y = 21 ()x@yII  x - 2y = -7  7y = 21 () FMicrosoft Equation 3.0 DS Eq_1060589539~F1˷01˷0Ole CompObj}fObjInfouation Equation.39q -2x - 3y + 5z = 15  4x -  y + 2z = -4 x  +  y - 3z = -7()@yII -2x - 3y + 5z = 15  Ole10NativeEquation Native _1060589538|F1˷01˷0Ole 4x -  y + 2z = -4 x  +  y - 3z = -7() FMicrosoft Equation 3.0 DS Equation Equation.39q  -2-3515  CompObjfObjInfoOle10NativeEquation Native 4-12-4  11-3-7 []ߠ@yII  -2-3515  4-12-4  11-3-7 [] FMicrosoft Equation 3.0 DS Eq_1060589537F1˷01˷0Ole CompObjfObjInfo !$%&(),/013458;<=?@CFGHJKLMPSTUWX[^_`bcfijklmnpqrsvyz{}uation Equation.39q  11-3-7  4-12-4  -2-3515 []ߠ@yII  11-3-7  4-12-4 Ole10NativeEquation Native _1060589536vF1˷01˷0Ole   -2-3515 [] FMicrosoft Equation 3.0 DS Equation Equation.39q  11-3-7  0-51424  0-1-11 []CompObj fObjInfo Ole10Native Equation Native ߠ@yII  11-3-7  0-51424  0-1-11 [] FMicrosoft Equation 3.0 DS Equation Equation.39q_1060589535F1˷01˷0Ole CompObjfObjInfoOle10NativeEquation Native _1060589534F1˷01˷0Ole   11-3-7  0-1-11  0-51424 []ߠ@yII  11-3-7  0-1-11  0-51424 []CompObj fObjInfo"Ole10Native#Equation Native ' FMicrosoft Equation 3.0 DS Equation Equation.39q  11-3-7  0-1-11  001919[]ߘ@yII  11-3-7  0-1-11  001919[] FMicrosoft Equation 3.0 DS Equation Equation.39q  x + y - 3z = -7 - y -  z = 119z =_1060589533F@Y˷0@Y˷0Ole *CompObj+fObjInfo-Ole10Native.Equation Native 2_1060589532F@Y˷0@Y˷0Ole 6 19()߸@yII  x + y - 3z = -7 - y -  z = 119z = 19() FMicrosoft Equation 3.0 DS EqCompObj7fObjInfo9Ole10Native:Equation Native >     4 "!#$%&'(*)+,.-0/12356789;:<=>?@BADCEFHGIJKLMNQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~uation Equation.39q  x - 3y = -18  2x + 5y =  19 ()ߔ@yII  x - 3y = -18  2x + 5y =  19 () FMicrosoft Equation 3.0 DS Equation Equation.39q  x -  y + 2z = 9  2x + 3y -  z = -2  x - 6y - 3z = -1 (_1060589531F@Y˷0@Y˷0Ole ACompObjBfObjInfoDOle10NativeEEquation Native I_1060589530F@Y˷0@Y˷0Ole N)@yII  x -  y + 2z = 9  2x + 3y -  z = -2  x - 6y - 3z = -1 () FMicrosoft Equation 3.0 DS EqCompObjOfObjInfoQOle10NativeREquation Native Vuation Equation.39q  * 1 a  * 1 b * 2 a  * 2 b[]ߔ@yII  * 1 a  * 1 b * 2 a  * 2 b[] FMicrosoft Equation 3.0 DS Equation Equation.39q  * 1 a  * 1 b * 2 a  * 2 bab_1060589529F@Y˷0@Y˷0Ole YCompObjZfObjInfo\Ole10Native]Equation Native a_1060589528^F@Y˷0@Y˷0Ole dߐ@yII  * 1 a  * 1 b * 2 a  * 2 b FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjefObjInfogOle10NativehEquation Native o  * 1 a  * 1 b * 2 a  * 2 bab = * 1 a* 2 b - * 2 a* 1 b@yII  * 1 a  * 1 b * 2 a  * 2 b = * 1 a* 2 b - * 2 a* 1 b FMicrosoft Equation 3.0 DS Equation Equation.39q_1060589527F@Y˷0`˷0Ole tCompObjufObjInfowOle10NativexEquation Native ||_1060589526F`˷0`˷0Ole ~ -5   3  2  -1 []`@yII -5   3  2  -1 [] FMicrosoft Equation 3.0 DS EqCompObjfObjInfoOle10NativeEquation Native uation Equation.39q  -5   3  2  -1ab = (-5 ) (-1) - (     ) (     )@yII  -5   3  2  -1 = (-5 ) (-1) - (     ) (     ) FMicrosoft Equation 3.0 DS Equation Equation.39q  3  -4 5  -6ab = 3 (-6) - (    )_1060589525F`˷0`˷0Ole CompObjfObjInfoOle10NativeEquation Native _1060589524F`˷0`˷0Ole  (    )߼@yII  3  -4 5  -6 = 3 (-6) - (    ) (    ) FMicrosoft Equation 3.0 DS EqCompObjfObjInfoOle10NativeEquation Native |uation Equation.39q -7   2  3  -4 ab`@yII -7   2  3  -4 _1060589523F`˷0`˷0Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q   12     15    5  -12 abOle10NativeEquation Native _1060589522F`˷0`˷0Ole ߄@yII   12     15    5  -12  FMicrosoft Equation 3.0 DS Equation Equation.39q * 1 ax + * 1CompObjfObjInfoOle10NativeEquation Native  by = * 1 c* 2 ax + * 2 by = * 2 c() with * 1 a* 1 b - * 2 a* 2 b _ 0l@yII * 1 ax + * 1 by = * 1 c* 2 ax + * 2 by = * 2 c() with * 1 a* 1 b - * 2 a* 2 b _ 0 FMicrosoft Equation 3.0 DS Equation Equation.39q_1060589521F "˷0 "˷0Ole CompObjfObjInfo  * 1 a  * 1 b * 2 a  * 2 babߐ@yII  * 1 a  * 1 b * 2 a  * 2 bOle10NativeEquation Native _1060589520F "˷0 "˷0Ole  FMicrosoft Equation 3.0 DS Equation Equation.39q  * 1 c  * 1 b * 2 c  * 2 babCompObjfObjInfoOle10NativeEquation Native ߐ@yII  * 1 c  * 1 b * 2 c  * 2 b FMicrosoft Equation 3.0 DS Equation Equation.39q  * 1 a  * 1 _1060589519F "˷0 "˷0Ole CompObjfObjInfoOle10NativeEquation Native _1060589518F "˷0 "˷0Ole c * 2 a  * 2 cabߐ@yII  * 1 a  * 1 c * 2 a  * 2 c FMicrosoft Equation 3.0 DS EqCompObjfObjInfoOle10NativeEquation Native uation Equation.39q x = * x DD  and   y = * y DDߐ@yII x = * x DD  and   y = * y DD FMicrosoft Equation 3.0 DS Equation Equation.39q 5x - 6y = -43x + 2y = -8()_1060589517F "˷0 ˷0Ole CompObjfObjInfoOle10NativeEquation Native _1060589516F ˷0 ˷0Ole ߄@yII 5x - 6y = -43x + 2y = -8() FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj fObjInfoOle10Native  Equation Native  D =  5  -6 3   2ab = 5(2) - (    )(    )@yII D =  5  -6 3   2 = 5(2) - (    )(    )_1060589515F ˷0 ˷0Ole CompObj fObjInfo    "#$%&')*+,-034578;>?@BCDGJKLMNOQRSTWZ[\^_`adghijklnopqruxyz|} FMicrosoft Equation 3.0 DS Equation Equation.39q * x D =   -4  -6  -8   2 ab = -4(2) - (    ) (    )Ole10NativeEquation Native _1060589514 F ˷0 ˷0Ole  @yII * x D =   -4  -6  -8   2  = -4(2) - (    ) (    ) FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoOle10NativeEquation Native $ * y D =  5  -4 3  -8ab = (     ) (     ) - (     ) (     )@yII * y D =  5  -4 3  -8 = (     ) (     ) - (     ) (     ) FMicrosoft Equation 3.0 DS Equation Equation.39q_1060589513F ˷0 ˷0Ole CompObjfObjInfo  x = * x DD = (      )(      ) =          y = * y DD = (      )(      ) =Ole10Native!Equation Native (l_1060589512N F ˷0 ˷0Ole .P@yII x = * x DD = (      )(      ) =          y = * y DD = (      )(      ) = FMicrosoft Equation 3.0 DS EqCompObj"/fObjInfo1Ole10Native!#2Equation Native 6uation Equation.39q 3x - 4y =  9   x - 4y = -1 ()ߐ@yII 3x - 4y =  9   x - 4y = -1 () FMicrosoft Equation 3.0 DS Equation Equation.39q D =  3  -4 1  -4ab = 3(-4) - (     ) (     )_1060589511&F˷0˷0Ole 9CompObj%(:fObjInfo<Ole10Native')=Equation Native A_10605895100$,F˷0˷0Ole E@yII D =  3  -4 1  -4 = 3(-4) - (     ) (     ) FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj+.FfObjInfoHOle10Native-/IEquation Native P< * x D =       -4        -4 ab = (    ) (     ) - (     ) (     ) @yII * x D =       -4        -4  = (    ) (     ) - (     ) (     ) FMicrosoft Equation 3.0 DS Equation Equation.39q_10605895092F˷0˷0Ole UCompObj14VfObjInfoX * y D =  3         1        ab = 3(    ) - 1(     )@yII * y D =  3         1         = 3Ole10Native35YEquation Native ]_1060589508B*8F˷0˷0Ole b(    ) - 1(     ) FMicrosoft Equation 3.0 DS Equation Equation.39q x = * x DD = (    )(    ) =    CompObj7:cfObjInfoeOle10Native9;fEquation Native mT        y = * y DD = (    )(    ) =8@yII x = * x DD = (    )(    ) =            y = * y DD = (    )(    ) = FMicrosoft Equation 3.0 DS Equation Equation.39q 2x + 3y = 11e cx]JAgf/ WhB`e"&D8P񌐀.6W[Zaa j%Bezr3Eq\ER#!r)MSp3+Yuƺ2 (h1? *eؔ)e5eiZy0oR|fvNU-o[uW|e|M6Q7kmoF;5SeunxQD3KsCAlf(b|"V㿊9a+nyU{dￃS9AØ, >; SKޯ3f@.($ qF.>"!Z7aDd<   C A? 2DoI\| 14۠j`!DoI\| 14۠j sxU=KAgf/C@FLK-L p!g bR ARYX 2h@ȹ3{dXnyo_f!@ $G^(($S--uyyLQG; ؆EP<>H&5h7Þ0?-B# . >ig<%1ߓe<4cU`z˞/7!ZYOl^~tKgݠu pktnfi55E$s[oKxS::WU~v-Oݿ@1vRµ%?%q4ƶhUEO|$wXXG`dePzn&)L afvyRLje>gDd<  C A ?  21#] (g"`!1#] (g `yxU;KAgv/\0X)l4,KS<H X&X  _[2/r^6;T|3Kk('mI'Gd>qEdVdH2 JXCA_q}zK{\y'5铢Y?V~=6v|rtI>b1(_wISOoP,vexfŀ0H*?a2ɋzįI3a8;\NJ4+?/˝OQpl7IAB85v==4I/uf\Ddx<  C A ?  2DS<@F5p?lL`!DS<@F5p?lL nxUKPO0:h - 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/2 G%J58I@p`!G%J58I@#x;KAg.0gb"| v  -$TJ; ?BZIii!"V6๳qasov,@{CqOlb Y%=s8!= B!ִ7 `qBw v' '~q\Y ^i16=F9hlE)З7}j~g A\KmORG!@&#w!pCB`cLMrr3=0G9Se2ì3^ 1G74c*x VPeTG8Ku%OU/GyJꚇdϡ._ymzѻ ONJS'k`OX3it*k7%ϔt BNV/WSk3IZ^? pnDd< 5 C A/? 027cmEs`!7cm@&xS/Q;v]Xwg DA@ ChV-٨HP@JJD*z{&x۝fCp,:@IDZbF[1ڻ.C6d5n^<ª]xFd`vXe-IB>qyD `.e1ܱgs;g's,6R 6`ǟ_/.' 4}#}+VCj|OjԚ&FM\Y>oH,k;v4+4xmv|r3AUM`gns {oc>BFW,[;>~uA9sqhkk/9L%dQ6x@ONLw읇c>Q3TbMhy܆"kR[/|k%x`/Ϣ)V׳ ^ER 9ęďR,n?<07i57nM:rt<ϸ"G1]))[Y5vnJYd2f{75)[gYc9=mǛ&Xgury7bDt~2[βDd'< : C A4? _1060589507>F˷0˷0Ole sCompObj=@tfObjInfovOle10Native?AwEquation Native {_1060589506H<DFU̷0U̷0Ole ~3x - 2y = -3()߈@yII 2x + 3y = 113x - 2y = -3() FMicrosoft Equation 3.0 DS EqCompObjCFfObjInfoOle10NativeEGEquation Native uation Equation.39q  x - 2y = -122x + 9y = 2  ()ߐ@yII  x - 2y = -122x + 9y = 2  () FMicrosoft Equation 3.0 DS Equation Equation.39q   * 1 a  * 1 b  * 1 c * 2 a  * 2 b  * 2 c* 3 a_1060589505JFU̷0U̷0Ole CompObjILfObjInfoOle10NativeKM$Equation Native _1060589504~6PFU̷0U̷0Ole   * 3 b  * 3 cab = * 1 a* 2 b* 3 c + * 1 b* 2 c* 3 a + * 1 c* 2 a* 3 b - * 3 a* 2 b* 1 c - * 3 b* 2 c* 1 a - * 3 c* 2 a* 1 b@yII  * 1 a  * 1 b  * 1 c * 2 a  * 2 b  * 2 c* 3 a  * 3 b  * 3 c = * 1 a* 2 b* 3 c + * 1 b* 2 c* 3 a + * 1 c* 2 a* 3 b - * 3 a* 2 b* 1 c - * 3 b* 2 c* 1 a - * 3 c* 2 a* 1 b FMicrosoft Equation 3.0 DS EqCompObjORfObjInfoOle10NativeQSEquation Native uation Equation.39q  2  -1   3 5   7  -41   6  -2abߜ@yII  2  -1   3 5   7  -41   6  -2 FMicrosoft Equation 3.0 DS Equation Equation.39q 2  7  -4 6  -2ab_1060589503VFU̷0U̷0Ole CompObjUXfObjInfoOle10NativeWYEquation Native |_1060589502`T\FU̷0U̷0Ole `@yII 2  7  -4 6  -2 FMicrosoft Equation 3.0 DS Equation Equation.39q -5  -1   3 6CompObj[^fObjInfoOle10Native]_Equation Native   -2abh@yII -5  -1   3 6  -2 FMicrosoft Equation 3.0 DS Eq_1060589501bFU̷0U̷0Ole CompObjadfObjInfouation Equation.39q + 1  -1   3   7  -4 abx@yII + 1  -1   3   7  -4Ole10NativeceEquation Native _1060589500rZhF ̷0 ̷0Ole   FMicrosoft Equation 3.0 DS Equation Equation.39q  2  -1   3 5   7  -41   6  -2abCompObjgjfObjInfoOle10NativeikEquation Native ߜ@yII  2  -1   3 5   7  -41   6  -2 FMicrosoft Equation 3.0 DS Equation Equation.39q_1060589499nF ̷0 ̷0Ole CompObjmpfObjInfoOle10NativeoqEquation Native |_1060589498xltF ̷0 ̷0Ole  2  7  -4 6  -2ab`@yII 2  7  -4 6  -2 FMicrosoft Equation 3.0 DS EqCompObjsvfObjInfoOle10NativeuwEquation Native uation Equation.39q - 5  -1   3 6  -2abl@yII - 5  -1   3 6  -2 FMicrosoft Equation 3.0 DS Equation Equation.39q + 1  -1   3   7  -4 ab_1060589497zF ̷0 ̷0Ole CompObjy|fObjInfoOle10Native{}Equation Native _1060589496fF̷0̷0Ole x@yII + 1  -1   3   7  -4  FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoOle10NativeEquation Native    !"#$%&'(*+,-./012369:;<=>?@ACDEFGHIJKLORSTVWX[^_`bcdehklmnoprstuvwz}~  1  -3   2   0   5  -4 6  -1   3abߨ@yII  1  -3   2   0   5  -4 6  -1   3_1060589495F̷0̷0Ole CompObj fObjInfo  FMicrosoft Equation 3.0 DS Equation Equation.39q * 1 ax + * 1 by + * 1 cz= * 1 d* 2 ax + * 2 by + * 2 cz= * 2 d*Ole10Native Equation Native _1060589494F̷0̷0Ole  3 ax + * 3 by + * 3 cz= * 3 d()߰@yII * 1 ax + * 1 by + * 1 cz= * 1 d* 2 ax + * 2 by + * 2 cz= * 2 d* 3 ax + * 3 by + * 3 cz= * 3 d() FMicrosoft Equation 3.0 DS Equation Equation.39qX D =  * 1 a*CompObjfObjInfoOle10Native\Equation Native ) 1 b* 1 c  * 2 a* 2 b* 2 c  * 3 a* 3 b* 3 cab _ 0,       * x D =  * 1 d* 1 b* 1 c  * 2 d* 2 b* 2 c  * 3 d* 3 b * 3 c ab ,p@yII D =  * 1 a* 1 b* 1 c  * 2 a* 2 b* 2 c  * 3 a* 3 b* 3 c _ 0,       * x D =  * 1 d* 1 b* 1 c  * 2 d* 2 b* 2 c  * 3 d* 3 b * 3 c  , FMicrosoft Equation 3.0 DS Equation Equation.39q_1060589493F̷0̷0Ole 4CompObj5fObjInfo7Ole10Native8\Equation Native B_1060589492F̷0̷0Ole MX * y D =  * 1 a* 1 d* 1 c  * 2 a* 2 d* 2 c  * 3 a* 3 d* 3 c ab ,          * z D =  * 1 a* 1 b* 1 d  * 2 a* 2 b* 2 d  * 3 a* 3 b* 3 d ab ,߄@yII * y D =  * 1 a* 1 d* 1 c  * 2 a* 2 d* 2 c  * 3 a* 3 d* 3 c  ,          * z D =  * 1 a* 1 b* 1 d  * 2 a* 2 b* 2 d  * 3 a* 3 b* 3 d  , FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjNfObjInfoPOle10NativeQEquation Native U x = * x DD ,   y = * y DD ,  and   z = * z DD.@yII x = * x DD ,   y = * y DD ,  and   z = * z DD. FMicrosoft Equation 3.0 DS Equation Equation.39q x +  y -  z = -22x - 3y + 4z = 17-3x + 2y + 5z = -7()_1060589491F̷0̷0Ole YCompObjZfObjInfo\Ole10Native]Equation Native a _1060589490F̷0̷0Ole f@yII x +  y -  z = -22x - 3y + 4z = 17-3x + 2y + 5z = -7() FMicrosoft Equation 3.0 DS EqCompObjgfObjInfoiOle10NativejEquation Native quation Equation.39q D =  11-1  2-34  -325 ab = 1  -34  25 ab - 2  1-1  25 ab + (-3)  1-1  -34 abߤ@yII D =  11-1  2-34  -325  = 1  -34  25  - 2  1-1  25  + (-3)  1-1  -34 _1060589489F̷0̷0Ole xCompObjyfObjInfo{ FMicrosoft Equation 3.0 DS Equation Equation.39q * x D =  -21-1  17-34  -725 ab = -2  -34  25 ab -17  1-Ole10Native|Equation Native _1060589488F'̷0'̷0Ole 1  25 ab + (-7)  1-1  -3 4 ab@yII * x D =  -21-1  17-34  -725  = -2  -34  25  -17  1-1  25  + (-7)  1-1  -3 4  FMicrosoft Equation 3.0 DS Equation Equation.39qX * y D =  1-CompObjfObjInfoOle10Native\Equation Native 42-1  2174  -3-75 ab = 1  174  -75 ab - 2  (   )(   )  -75 ab + (-3)  (   )(   )  (   )(   ) ab@yII * y D =  1-2-1  2174  -3-75  = 1  174  -75  - 2  (   )(   )  -75  + (-3)  (   )(   )  (   )(   )  FMicrosoft Equation 3.0 DS Equation Equation.39qX * z D =  11-2  2-317  -32-7_1060589487F'̷0'̷0Ole CompObjfObjInfoOle10Native\Equation Native 8_1060589486F'̷0'̷0Ole  ab = 1  -317  2-7 ab - 2  1-2  (   )(   ) ab + (-3)  (   )(   )  (   )(   ) ab@yII * z D =  11-2  2-317  -32-7  = 1  -317  2-7  - 2  1-2  (   )(   )  + (-3)  (   )(   )  (   )(   )  FMicrosoft Equation 3.0 DS Equation Equation.39qX x = * x DD = (    )(    ) =        ;  y = * y DDCompObjfObjInfoOle10Native\Equation Native  = (    )(    ) =        ;  z = * z DD =  (    ) (    ) =    @yII x = * x DD = (    )(    ) =        ;  y = * y DD = (    )(    ) =        ;  z = * z DD =  (    ) (    ) =    _1060589485F'̷0'̷0Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q 2x -  y +  z = 0 x - 3y + 4z = 10 5x + 2y - 3z = -14()Ole10NativeEquation Native _1060589484F'̷0'̷0Ole @yII 2x -  y +  z = 0 x - 3y + 4z = 10 5x + 2y - 3z = -14() FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoOle10NativeEquation Native   y = * 2 x - 4  y = 2x - 1 ()ߐ@yII  y = * 2 x - 4  y = 2x - 1 ()_1060589483F/̷0/̷0Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q y = * 2 x - 4y = 2x - 1Ole10NativeEquation Native _1060589482F/̷0/̷0Ole p@yII y = * 2 x - 4y = 2x - 1 FMicrosoft Equation 3.0 DS Equation Equation.39q  * 2 x + * 2 CompObjfObjInfoOle10NativeEquation Native y = 4 y = * 2 x - 4()ߤ@yII  * 2 x + * 2 y = 4 y = * 2 x - 4() FMicrosoft Equation 3.0 DS Eq_1060589481F/̷0/̷0Ole CompObjfObjInfo  !#uation Equation.39q y > 2x - 3 x - 3y  6 ()x@yII y > 2x - 3 x - 3y e" Ole10NativeEquation Native _1060589480F/̷0/̷0Ole  6 () FMicrosoft Equation 3.0 DS Equation Equation.39q y  32x - y < 3()CompObj fObjInfo Ole10Native Equation Native x\@yII y d" 32x - y < 3() Oh+'0x  4 @ LX`hpWorksheet 54 (10dorkTL UserL UL U Normal.dot4TL User      !"#52"Ti `!Ti @@%xS/QcBAB(W\B8?.UK6:h@P)D% P!֛U&ٻff<@[R=)bQ!RE:2&4Mdyw Y5BDLOkU^cr20T^^l^֭+b'^Ny%mԵ#ƶ6>?qC8Jw~Ge:9w?v0 &5ga|g_qFa(!3BNr5feOY10/ZШ'ZJ oA33ߥҦl޳+uN]W+gP?dXQ2ZMʳ9IGDcʓAƻowH]6+]JZ5gk4*\طc98hDc+tudw,e^PTOB 2B2aHڌ Գɓ?\gDdH< < C A6? 72ohf2Oׄ`!ohf2O @%"yx]JAg A%$,_@ N"x*$\o y$EJRZ`cT0'aAT£ l|YYZ̏pq+},D:>R q!{ KVPo!弢!VX2D(u{<ǁ)x!ޕͦ!tVafwiS; a3YDdH< = C A7? 82).BJlVj>`!).BJlVj@%9e kxMhAߛfeiZx^"jKR T1~Pۛқ,ҫ" CA8+x(1&{raofDِCä}B fâꞓ<{qяK)$ erum5*5L7IJ}yqfZ"ԓ3٣<֐hs~_ oqZd-^UW Z:цGf1G~=K^+]uҗ{,mm__Һk2 ϑ\p|E~)|3| (sb$8Y~H#F~")Շ0$A+ + `>ƑG| y{g)ȟ +=o0nAiHU(ME_HN(w[1&p\ogo㯒*o ]?J|&S1S>+su%?=O9?Ϡt}aԳ)$`oC(E5ByB_rPrη˥-t; C A8? 92 Ɩ%A?`! 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