Systems of Linear Equations Introduction

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1 Systems of Linear Equations Introduction Linear Equation a x = b Solution: Case a 0, then x = b (one solution) a Case 2 a = 0, b 0, then x (no solutions) Case 3 a = 0, b = 0, then x R (infinitely many solutions, moreover, any x R is a solution) Geometrical Interpretation Solution is the x-intercept of the graph of the function y = a x b Case Slope = a 0 - one intersection Case 2 Slope = a = 0, b 0 - the graph is parallel to x axes - no intersection Case 3 Slope = a = 0, b = 0 - the graph coincides with x axes - infinitely many intersections 2 System of Linear Equations 2 Substitution Method a x + y = c a 2 x + y = c 2 Suppose 0, then from the first equation y = c a x Substituting to the second we obtain one variable equation a 2 x + c a x = c 2

2 Assign to a system a x + y = c 22 Elimination Method Multiply the first equation by a 2 a and add to the second a x + y = c a x + y = c a 2 x + y = c 2 (a 2 a 2 a a ) x + ( a 2 a )y = c 2 a 2 a c, a x + y = c ( a 2 a )y = c 2 a 2 a c Thus y = c 2 a 2c a a 2 = a c 2 a 2 c a a a 2 and the back substitution gives 23 Determinant Method x = c c 2 a a 2 three DETERMINANTS = x = c c 2 a 2 x + y = c 2 a a 2 = a a 2, = c c 2, y = a c a 2 c 2 = a c 2 a 2 c 23 Cramer s Rule Case If 0 then x = x, y = y Case 2 If = 0 and either x 0 or y 0 then the system has NO SOLUTIONS Case 3 If = 0 and x = 0, y = 0 then the system has INFINITELY MANY SOLUTIONS Example Try to solve the system x + 0 y = c for various c and c 2 x + 0 y = c 2 2

3 24 Geometrical Interpretation Each equation of the system defines linear function y = a x + c, y = a 2 x + c 2 A solution of the system is the intersection point of their graphs Case These two graphs have different slopes thus they have one intersection point: = a a 2 0 a a 2 a a 2 Case 2 These two graphs have equal slopes but different y-intercepts thus they are parallel: but = a a 2 = 0 a = a 2 a = a 2, x = c c 2 0 c c 2 c c 2 Case 3 These two graphs have equal slopes and the same y-intercepts thus they coincide: = a a 2 = 0 a = a 2 a = a 2, but x = c c 2 = 0 c = c 2 c = c 2 25 More About the Case 3 In this case the system has infinitely many solution but not all the couples (x, y) form a solution! = a a 2 = 0, means that a = a 2 thus a a 2 = Similarly x = c c 2 = 0 c = c 2 c c 2 = So in this case a a 2 = b2 = c c 2 b2 b2 ie the equations are proportional a x + y = c ka x + k y = kc and any solution of the first equation satisfies the second one The couples (x, y = c a x ) form all the solutions 3

4 3 General form of m linear equations with n variables Three ingredients of the system A = a x + a 2 x a n x n = c a 2 x + a 22 x a 2n x n = c 2 a m x + a m2 x a mn x n = c m a a 2 a n a 2 a 22 a 2n a m a m2 a mn x = x x 2 x n c = A coefficient matrix of the system of order m n, x vector of variables of order n, c vector of constants of order m A and C together form so called augmented matrix A = a a 2 a n c a 2 a 22 a 2n c 2 a m a m2 a mn c m c c 2 c m The method of elimination of variables described bellow uses the following elementary equation operations: Adding a multiple of one equation to another 2 Multiplying both sides of an equation by nonzero scalar 3 Interchanging of two equations If a system is obtained from another one by these operations, they are equivalent, that is they have same solutions To these three elementary equation operations correspond three elementary row operations on the augmented matrix: Adding a multiple of one row to another 2 Multiplying each element of a row by the same nonzero scalar 3 Interchanging of two rows Definition A row of a matrix is said to have k leading zeros if first k elements of the row are all zeros 4

5 Definition A matrix is in row echelon form if each row has more leading zeros that the row preceding it The first non-zero entry in each row is called a pivot Definition A row echelon matrix is in reduced row echelon form if each pivot is and each column containing a pivot has no other nonzero entries Example The matrix A is in row echelon form, and B is in reduced row echelon form A = , B = Every non-zero matrix can be reduced to an infinite number of row echelon forms (they can all be multiples of each other, for example) via elementary matrix transformations 3 Gaussian Elimination Step Forward elimination: Choose a pivot element on the main diagonal (start at top left corner) Eliminate (set to zero) all elements below the main diagonal Move to the next column Repeat until matrix is in row echelon form Step 2 Solve the unknowns using backward substitution If necessary introduce free variables and solve basic variables 32 Gaussian-Jordan Elimination This method reduces the matrix even further to reduced row echelon form For this: Choose a pivot element on the main diagonal (start at top left corner) Eliminate (set to zero) all elements above and below the main diagonal Move to the next column 5

6 Another way: suppose we have a matrix already in row echelon form Then first convert each pivot a to multiplying its row by, then use these a pivots (starting with the in the last row) to kill all the entries above it 32 Example of a system with one solution Consider the system 2x + y z = 8 3x y + 2z = 2x + y + 2z = 3 The augmented matrix of this system is The Gauss elimination algorithm gives I + II I + III II + III This is the row echelon form So our system is equivalent to the following system 2x + y z = 8 2 y + 2 z = z = 6

7 and x, y, z can be found by back substitution: x = 2, y = 3, z = The Gauss-Jordan elimination algorithm gives the reduced row echelon form So the solution is the last column x = 2, y = 3, z = 322 Example of a system with many solutions Consider the system x + 2y + 0 z + 3t = 5 2x + 3y + 2z + 5t = 4 0 x + 2y + z + 5t = 2 The augmented matrix of this system is The Gauss elimination algorithm gives the following row echelon form Corresponding system looks as x + 2y + 0 z + 3t = 5 y + 2z t = 6 5z + 3t = 0 Take free variable t Then the back substitution gives z = 3 5 t 2, y = 5 t + 2, x = 7 5 t + 7

8 323 Example of a system with no solutions Consider the system 2x + 3y + z = x + y + 2z = 2 3x + 4y + 3z = 4 The augmented matrix of this system is The Gauss elimination algorithm gives the following row echelon form Corresponding system looks as x + 3y + z = 0 x y + 23z = 3 0 x + 0 y + 0 z = 4 so the system has no solutions, that is it is non consistent 4 Economical Examples 4 One-commodity market equilibrium Q d - demand of commodity, Q s - supply of commodity, P - price per unit Linear model Q d (P ) = kp + b (decreasing function), Q s (P ) = lp + c (increasing function) Equilibrium condition Q d (P ) = Q s (P ) Equilibrium price k P + b = l P + c, (l + k) P = b c, P = b c l + k 8

9 42 Two-commodity market equilibrium Q d demand of commodity ; Q s supply of commodity ; Q d2 demand of commodity 2, Q s2 supply of commodity 2; P price per unit of commodity ; P 2 price per unit of commodity 2 Demand function for commodity : Demand function for commodity 2: Supply function for commodity : Supply function for commodity 2: Q d (P, P 2 ) = a 0 + a P + a 2 P 2 Q d2 (P, P 2 ) = α 0 + α P + α 2 P 2 Q s (P, P 2 ) = b 0 + P + P 2 Q s2 (P, P 2 ) = β 0 + β P + β 2 P 2 Equilibrium conditions Q d = Q s, Q d2 = Q s2 : a0 + a P + a 2 P 2 = b 0 + P + P 2 α 0 + α P + α 2 P 2 = β 0 + β P + β 2 P 2 (a ) P + (a 2 ) P 2 = b 0 a 0 (α β ) P + (α 2 β 2 ) P 2 = β 0 α 0 Equilibrium prices P and P 2 can be solved from this system 43 Equilibrium in National Income I investment, G spending by government, a autonomous consumption, b marginal propensity to consumption, Y National Income, C consumption Equilibrium condition Y = C + I + G C = a + by Y C = I + G by C = a Y = I+G+a b C = a+b (I+G) b 9

10 Exercises Solve these systems using all 3 methods (Gauss, Gauss-Jordan, determinant), and give the suitable graphical interpretation for 2x y = 5 x + y = 4 2 2x y = 5 4x 2y = 4 3 2x y = 5 4x 2y = 0 2 Give examples of systems 2 2 and 3 3 with (a) one solution, (b) no solutions, (c) infinitely many solutions 2x + ky = 8 3 Find the values of k and c for which the system is (a) 4x + 8y = c consistent (ie it has solutions), (b) inconsistent (ie it has no solutions) Exercises 7-73, from SB 0

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