MAT223H1a - MAT223H1a.doc Page 1 of 38 Lecture #1 –...

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Unformatted text preview: MAT223H1a.doc Page 1 of 38 Lecture #1 – Tuesday, January 6, 2004 1.1 S OLUTIONS AND E LEMENTARY O PERATIONS Example Create a diet from fish and meal that contains 193g of proteins and 83g of carbohydrate. We know that fish contains 70% protein and 10% carbohydrate, and meal contains 30% protein and 60% carbohydrate. • Assume that the diet contains x g of fish and y g of meal, we obtain & ¡ ¢ = + = + 83 6 . 1 . 193 3 . 7 . y x y x . Definition A linear equation is an equation of the form b x a x a n n = + + ... 1 1 where: • x 1 ,…, x n are variables; • a 1 ,…, a n are real numbers called coefficients; • b is the constant term. Examples 1) c by ax = + . 2) 2 3 3 2 1 =- + x x x . 3) 1 2 2 2 1 = + x x – not a linear equation. Definition A finite collection of linear equations in the variables x 1 ,…, x n is called a system of linear equations . Examples 1) & ¡ ¢ = = 3 1 2 1 x x . 2) & ¡ ¢ =- = + 3 14 4 2 2 1 2 1 x x x x Definition Given a linear equation b x a x a n n = + + ... 1 1 , a sequence of n real numbers s 1 ,…, s n is called a solution to the linear equation if b s a s a n n = + + ... 1 1 . Similarly, this also applies to a given system of linear equations. Example Given ( ) & ¡ ¢ = = 3 1 2 1 1 x x S and ( ) & ¡ ¢ =- = + 3 14 4 2 2 1 2 1 2 x x x x S , what is the solution to ( S 1 ) and ( S 2 )? • (1, 3) is a solution of ( S 1 ). • (1, 3) is also a solution of ( S 2 ) because () ( ) 14 3 4 1 2 = + and () ( ) 3 1 3 =- . MAT223H1a.doc Page 2 of 38 Example • ( ) & ¡ ¢ = + = + 2 1 3 y x y x S has no solution. Example Prove that for any s , t in R, £ £ £ & £ £ £ ¡ ¢- = = =- + = s s t s s s t s s 2 2 1 2 3 4 3 2 1 is a solution to the system & ¡ ¢ = + + + =- +- 2 1 3 4 3 2 1 4 3 2 1 x x x x x x x x . • 1 3 2 1 2 3 2 2 1 3 2 3 3 4 3 2 1 = +- +- +- = ¤ ¥ ¦ § ¨ ©-- +-- + =- +- t t s s s s t s t s s s s s . • 2 2 2 1 2 3 2 2 1 2 3 4 3 2 1 = +-- + + + =- + + +- + = + + + t t s s s s t s t s s s s s . Definition s , t are called parameters . s 1 ,…, s 4 described this way is said to be given in parametric form and is called the general solution of the system. Remarks • When only 2 variables are involved, solution to systems of linear equations can be described geometrically because a linear equation c by ax L = + : is a straight line if a , b are not both 0. ( ) 2 1 , s s P is in L if it is a solution of c by ax = + . • If there are two linear equations, c by ax L = + : 1 and f ey dx L = + : 2 , then the solution to the system ( ) & ¡ ¢ = + = + f ey dx c by ax S is the intersection of L 1 and L 2 . • The solution of (S) is ( ) 2 1 , s s where ( ) 2 1 2 1 , L L s s P ∩ = . MAT223H1a.doc Page 3 of 38 • The solution of ( S ) are given by the ( ) 2 1 , s s such that c bs as = + 2 1 (and this implies that f es ds = + 2 1 )....
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This note was uploaded on 10/12/2011 for the course MAT 223 taught by Professor Uppa during the Fall '09 term at University of Toronto.

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MAT223H1a - MAT223H1a.doc Page 1 of 38 Lecture #1 –...

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