Systems of Equations

# Systems of Equations - Systems of Equations Lets start off...

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Systems of Equations Let’s start off this section with the definition of a linear equation . Here are a couple of examples of linear equations. In the second equation note the use of the subscripts on the variables. This is a common notational device that will be used fairly extensively here. It is especially useful when we get into the general case(s) and we won’t know how many variables (often called unknowns) there are in the equation. So, just what makes these two equations linear? There are several main points to notice. First, the unknowns only appear to the first power and there aren’t any unknowns in the denominator of a fraction. Also notice that there are no products and/or quotients of unknowns. All of these ideas are required in order for an equation to be a linear equation. Unknowns only occur in numerators, they are only to the first power and there are no products or quotients of unknowns. The most general linear equation is, (1) where there are n unknowns, , and are all known numbers. Next we need to take a look at the solution set of a single linear equation. A solution set (or often just solution) for (1) is a set of numbers so that if we set , , … , then (1) will be satisfied. By satisfied we mean that if we plug these numbers into the left side of (1) and do the arithmetic we will get b as an answer. The first thing to notice about the solution set to a single linear equation that contains at least two variables with non-zero coefficents is that we will have an infinite number of solutions. We will also see that while there are infinitely many possible solutions they are all related to each other in some way.

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Note that if there is one or less variables with non-zero coefficients then there will be a single solution or no solutions depending upon the value of b . Let’s find the solution sets for the two linear equations given at the start of this section. Example 1 Find the solution set for each of the following linear equations. (a) [ Solution ] (b) [ Solution ] Solution (a) The first thing that we’ll do here is solve the equation for one of the two unknowns. It doesn’t matter which one we solve for, but we’ll usually try to pick the one that will mean the least amount (or at least simpler) work. In this case it will probably be slightly easier to solve for so let’s do that. Now, what this tells us is that if we have a value for then we can determine a corresponding value for . Since we have a single linear equation there is nothing to restrict our choice of and so we we’ll let be any number. We will usually write this as , where t is any number. Note that there is nothing special about the t , this is just the letter that I usually use in these cases. Others often use s for this letter and, of course, you could choose it to
be just about anything as long as it’s not a letter representing one of the unknowns in the equation ( x in this case). Once we’ve “chosen”

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Systems of Equations - Systems of Equations Lets start off...

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