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Week_2 - Week 2 1.1 Systems of Linear Equations A linear...

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Week 2 1.1 Systems of Linear Equations A linear equation in the n unknowns x 1 , x 2 , . . . x n is an equation of the form a 1 x 1 + a 2 x 2 + . . . a n x n = b, where a 1 , a 2 , . . . a n , b are real constants. The graphs of linear equations in 2 variables are lines. ex. 2 x + y = 3 A solution of a linear equation is a sequence of numbers ( s 1 , s 2 , . . . s n ) so that the equation is satis- fied when we substitute x 1 = s 1 , x 2 = s 2 , . . . , x n = s n . ex. A solution to 2 x + y = 3 is ( - 1 , - 1). Geometrically, a solution is a point on the line. A system of linear equations is a set of linear equations. A solution to a linear system must satisfy all equations in the system. Every linear system has either: 1. No solutions 2. Exactly 1 solution 3. Infinitely many solutions Geometrically, a solution to a system of 2 linear equations in 2 unknowns is a point of intersection. To find the solution(s), if any, to a linear system, we can perform elementary operations on the equations such as multiplying both sides of an equation by a scalar, or adding/subtracting equations together to come up with a simpler system with the same number of equations.
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