Linear algebra involves the extension of techniques from elementary algebra to the solution of systems of linear equations. A linear equation is one in which no two variables are multiplied together, so that terms like xy, yz, x2, y2, and so on, do not appear. A system of equations is a set of two or more equations containing the same variables. Systems of equations arise in situations where two or more unknown quantities are present. In order for a unique solution to exist there must be as many independent conditions given as there unknowns, so that the number of equations that can be formulated equals the number of variables. Thus, we speak of two equations in two unknowns, three equations in three unknowns, and so forth. Consider the example of finding two numbers such that the first is six times larger than the second, and the second is 10 less that the first. This problem has two unknowns, and contains two independent conditions. In order to determine the two numbers, let x represent the first number and y represent the second number. Using the information provided, two equations can be formulated, x = 6y, from the first condition, and x−10 = y, from the second condition. To solve for y, replace x in the second equation with 6y from the first equation, giving 6y−10=y. Then, subtract y from both sides to obtain 5y−10=0, add 10 to both sides giving 5y=10, and divide both sides by 5 to find y=2. Finally, substitute y=2 into the first equation to obtain x=12. The first number, 12, is six times larger than the second, 2, and the second is 10 less than the first, as required. This simple example demonstrates the method of substitution. More general methods of solution involve the use of matrix algebra.
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