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MATHPRELIM

MATHPRELIM - Mathematical Preliminaries for ECOS2001...

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Mathematical Preliminaries for ECOS2001 Oleksii Birulin 2008 1 Linear equations linear equation always has a form ax + b = c; where a; b; c are some numbers and x stands for the variable that we have to solve for. the way to solve a linear equation: °rst subtract b from both sides± we get ax = c ° b now divide both sides by a and we get the answer x = c ° b a 1.1 Examples Suppose we have to solve 5 x + 8 = 10 take 8 from both sides we get 5 x = 2 now divide by 5 and we get an answer x = 2 5 Another example ° 3 y ° 10 = 23 add 10 to both sides (or equivalently take -10 from both sides) we get ° 3 y = 33 now divide by -3 we get an answer y = ° 11 1

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1.2 Additional exercises Solve the equations 2 x + 5 = 10 4 x ° 5 = 16 1 3 x + 1 = 0 ° 5 6 y ° 1 6 = 2 ° 5 6 y + 1 3 = 1 2 Systems of linear equations We deal with systems of 2 equations only. In general the system of 2 linear equations can be represented as ax + by + c = d ex + fy + g = h where a; b; c; d; e; f; g; h are some numbers and x; y are the variables that we are interested in. note that to pin down 2 variables simultaneously we have to have 2 equations, one is not enough. The easiest way to solve the system is use one of the equations to °gure out y in terms of x and then substitute the expression that you get into the remaining equation. What we obtain as a result of this substitution is a linear
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MATHPRELIM - Mathematical Preliminaries for ECOS2001...

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