1 Review of two equations in two unknowns


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1 Contents 1 Review of two equations in two unknowns 1.1 The "standard" method for finding the solution 1.2 The geometric method of finding the solution 2 Some equations for which the "standard" method doesn't work 2.1 Equations with no solutions 2.2 Equations with more than one solution 3 Important Definitions 3.1 The Definition of a Linear Equation 3.2 The Definition of a System of Linear Equations 3.3 The Coefficient and Augmented Matrix of a System of Linear Equations 1 Review of two equations in two unknowns 1.1 The "standard" method for finding the solution Suppose we want to find all solutions to the equations The "standard" technique is to manipulate one or both of the equations until either has the same coefficient in both equations or has the same coefficient, and then subtract to eliminate one variable. Since there is only one variable left, its value can be found; with this information, the value for the other variable can be found. In the case above, we can multiply both sides of the second equation by, to get, and subtracting from the first equation gives, or. Using the first equation and the (now) known value of, we find that. Hence we see that there is exactly one pair of values for and that simultaneously satisfy both equations: and. We can also write this as. In this case we say that there is a unique solution (in mathematics, the term unique means "exactly one"). 1 Review of two equations in two unknowns 1
2 1.2 The geometric method of finding the solution The set of equations solved in the previous section can also be viewed geometrically. The points in the  plane satisfying one of the equations lie on a straight line, and any point satisfying both of the equations must lie on both lines. The plot of the two lines looks like this: The point of intersection is, just as before. 2 Some equations for which the "standard" method doesn't work 2.1 Equations with no solutions Suppose we want to find all solutions to the equations Using the "standard" approach we multiply the second equation by variable. This leaves us with the equation and add it to the first one to eliminate the 2 Some equations for which the "standard" method doesn't work 2
3 This is certainly an equality that is not valid. What happened? We can see by multiplying the second equation by. We then have Whatever and are, the value of can't be and at the same time. So there are no solutions. What happens if we try to graph these two equations? Here is what we get: The geometry of the situation is now clear: the two lines are parallel and so there is no point on both lines (indeed, both lines have slope ). When we have equations with no common solution, they are called inconsistent. 2.2 Equations with more than one solution Now let's alter the situation of the previous section slightly. We consider the pair of equations We apply the "standard" method again: multiply the second equation by and add it to the first. The result is 2.1 Equations with no solutions 3
4 This is certainly a valid, although not very interesting, equation. In fact, if we multiply both sides of the second equation by the system becomes This means that any solution of the first equation is also a solution of the second one. Geometrically, if we plot the graph of the two equations, the same line results for each one. How do we find all solutions in this case? Let us assign a value to. Let's call it so. Then, using either equation, we have. This means that for any value of we know that is a solution to both equations. So, in fact we have an infinite number of solutions. 3 Important Definitions 3.1 The Definition of a Linear Equation The equations given above are linear equations in two variables. A linear equation in is an equation of the form variables where and are constants. The numbers are called the coefficients of the equation, that is, is the coefficient of, is the coefficient of, etc. This is also called a linear equation in unknowns. The following are examples of linear equations: (one variable) (two variables) (three variables) (four variables) (five variables: the ones with a coefficient of zero are not written) 3.2 The Definition of a System of Linear Equations A system of linear equations in unknowns is a list of linear equations, each of which has the same set of unknowns. They are usually presented with the purpose of finding all of their simultaneous solutions. The original example given above is a system of 2 equations in 2 unknowns. Here are some more examples: A system of 2 equations in 3 unknowns: 3 Important Definitions 4
5 A system of 3 equations in 2 unknowns: A system of 4 equations in 5 unknowns A system of equations in unknowns : 3.3 The Coefficient and Augmented Matrix of a System of Linear Equations A matrix is a rectangular array of numbers (the plural is matrices). Here is a matrix: This matrix has 3 rows: is row 1, is row 2 and is row 3. Similarly, the matrix has 4 columns: 3.2 The Definition of a System of Linear Equations 5
6 ,,, and are column 1, column 2, column 3 and column 4 respectively. We call a matrix with 3 rows and 4 columns a matrix. More generally, a matrix with rows and columns is called an matrix. An matrix has the form This notation means that is the number in both row and column. We will call the rows and the columns. In other words, and, The notation for this matrix is similar to that used for a system of linear equations, and for good reason. Suppose we have a system of linear equations in unknowns: The coefficient matrix is then the matrix 3.3 The Coefficient and Augmented Matrix of a System oflinear Equations 6
7 and the augmented matrix of the system is the matrix Hence the augmented is the coefficient matrix with one column (the constants on the right side of the equations) added. To emphasize the extra column, the augmented matrix is sometimes written as Here are some examples of coefficient matrices and augmented matrices: A system of 2 equations in 3 unknowns: Coefficient matrix: Augmented matrix: A system of 3 equations in 2 unknowns: Coefficient matrix: 3.3 The Coefficient and Augmented Matrix of a System oflinear Equations 7
8 Augmented matrix: A system of 4 equations in 5 unknowns Coefficient matrix: Augmented matrix: 3.3 The Coefficient and Augmented Matrix of a System oflinear Equations 8
2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
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