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1.1_1.2

# 1.1_1.2 - MATH1850U/2050U Chapter 1 1 LINEAR SYSTEMS...

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MATH1850U/2050U: Chapter 1 1 LINEAR SYSTEMS Introduction to Systems of Linear Equations (1.1; pg.2) Definition: A linear equation in the n variables, n x x x , , 2 1 is defined to be an equation that can be written in the form: b x a x a x a n n 2 2 1 1 where n a a a , , 2 1 and b are real constants. The i a are called the coefficients , and the variables i x are sometimes called the unknowns . If it cannot be written in this form, it is called a nonlinear equation . Examples: 4 3 2 1 8 10 6 5 x x x x 4 2 2 3 1 6 3 9 x x x x 9 7 3 2 1 x x x Application: Suppose that \$100 is invested in 3 stocks. If A , B, and C, denote the number of shares of each stock that are purchased and they have units costs \$5, \$1.5, and \$3 respectively, write the linear equation describing this scenario. Definition: A solution of a linear equation b x a x a x a n n 2 2 1 1 is a sequence (or n -tuple) of n numbers n s s s , , 2 1 such that the equation is satisfied when we substitute n n s x s x s x , , 2 2 1 1 in the equation. The set of ALL solutions of the equation is called the solution set (or sometimes the general solution ) of the equation. Example:

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MATH1850U/2050U: Chapter 1 2 Definition: A finite set of linear equations in n variables, n x x x , , 2 1 , is called a system of linear equations or a linear system . A sequence of n numbers n s s s , , 2 1 is a solution of the system of linear equations if n n s x s x s x , , 2 2 1 1 is a solution of every equation in the system. Example: Verify that 1 , 1 , 1 z y x is a solution of the linear system: 1 2 4 3 2 3 2 z y y x z y x Question: Does every system of equations have a solution? Definition:
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1.1_1.2 - MATH1850U/2050U Chapter 1 1 LINEAR SYSTEMS...

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