When only two variables are involved, the solutions to systems of linear equations can be described geometrically because the graph of a linear equation is a straight line if and are not both zero. Note that each variable in a linear equation occurs to the first power only. Is a linear equation the coefficients of, , and are, , and, and the constant term is. A finite collection of linear equations in the variables is called a system of linear equations in these variables.
Here denote real numbers (called the coefficients of, respectively) and is also a number (called the constant term of the equation).
Is called a linear equation in the variables. However, it is often convenient to write the variables as, particularly when more than two variables are involved. Is a straight line (if and are not both zero), so such an equation is called a linear equation in the variables and. If, , and are real numbers, the graph of an equation of the form Linear algebra arose from attempts to find systematic methods for solving these systems, so it is natural to begin this book by studying linear equations. Practical problems in many fields of study-such as biology, business, chemistry, computer science, economics, electronics, engineering, physics and the social sciences-can often be reduced to solving a system of linear equations. 1 System of Linear Equations 1.1 Solutions and elementary operations
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