Let's look at the graph of the second equation. Each equation can be graphed using an aspect drawing as a plane. Although an additional variable (z) is added, the concepts and method of solution are all the same. Let's now take a look at a system of equations in three variables. We can check this result by substituting these values back into the original equations. Now, we substitute this expression into the second equation:įinally, we substitute this value back into either expression to find the solution to the system of equations:
Let's use the example above to illustrate. This leaves an equation in one variable that can then be solved to find part of the solution.
We can solve just one of the equations for one variable or the other (usually whichever is easiest) and then use substitution to eliminate one variable from the other equation. Solving the system of equations expressed in this manner is essentially the same, but we give it a slight twist. The following, however, is an equivalent expression of the same system of equations. This is actually a simple example of a system of linear equations.īecause these functions are expressed in a relatively simple form (the variable y is isolated on one side of the equation in both cases), the method of solution is also simple. Let's briefly look at an example again, but let's write it using our x- y- z notation. In lesson 4, we solved some problems that involved two functions by equating the expressions. The conventional arrangement of axes for a graph of a function in three dimensions is shown below. For the purposes of this section, we will only use variables x, y, and z to represent the three dimensions. Additional dimensions, however, become extremely difficult to visualize (spatially, our experience of the world is in three dimensions only). We can plot such functions using a planar graph if a third dimension is added, we can still use aspect drawings to create the illusion of a three-dimensional graph. So far, we have only dealt with two-dimensional equations: those that have an independent variable (one dimension) and that have a dependent variable (another dimension). The area of mathematics that deals specifically with this type of problem is called linear algebra, which is a subject to which we could devote a course or two of its own! Because of its complexity, we will not deal with many of the aspects of linear algebra, but we will briefly cover what constitutes a system of linear equations and one reliable method for solving them.Ī system of linear equations can be expressed in terms of multiple variables. A system of linear equations is a set of equations (in some number of variables that may be greater than one or two) that must all be solved simultaneously. Likewise, we can also solve for the intersection (if it exists) of many linear functions in multiple dimensions by analyzing the associated system of linear equations. By solving this expression, we can obtain the solution (intersection). We can find the intersection of two lines by setting equal the functions that describe the lines. O Solve systems of linear equations in multiple dimensions
O Draw graphs of expressions that represent planes O Recognize equations in multiple dimensions (beyond just two)
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