To Be or Not To Be a Linear Equation: That Is the Question


 Dana McCoy
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1 To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not both 0. This form is called standard form. In standard form the variables and are on the same side of the equals sign, A is positive, and an constant is on the other side of the equal sign. A line contains an infinite number of points and each ordered pair is a solution of its corresponding equation. To determine if an equation is linear or not, we look at the eponents. If the highest eponent on a single variable is one, then the equation is a linear equation. A single variable means that the variables are not multiplied. Here is the standard form of a linear equation in two variables again. A + B C Look at the. The eponent is understood to be 1. Look at the. The eponent is understood to be 1. Eample 1: Let s look at several eamples of linear equations, and some that are not linear equations. is a linear equation since the eponents of and are both one. + is a linear equation since the eponents of and are both one. However, the equation is not written in standard form. We will learn how to change this equation to standard form later is not a linear equation since is in the denominator of the fraction is a linear equation since the eponents of and are both one. is not a linear equation since the variables and are being multiplied together. is not a linear equation since the eponent of is not one.
2 Remember, in order for an equation to be a linear, both variables need to have an eponent of one. Usuall, ou will not see the eponent of one written, but we know that it is there. An equation cannot be linear if an of the variables are in the denominator, multiplied together, squared, or under a radical. Linear or NonLinear Quick Check: Now ou tr some eamples! Determine if the following equations are linear or nonlinear. Circle our answer and describe wh linear nonlinear 4 Graphs of Equations A graph is a picture of the relationship between two variables, in our case, the variables and. The graph of a linear equation is a line. To find a relationship between two variables we need to first find ordered pairs, or points, (,), that make the equation true. These are often arranged in what we call a ttable.
3 Eample : If we had the equation + 8, we first create a ttable. Under the we place a. This means to replace b and find the corresponding value. We can arrange this information in a ttable. Now, select other values for or for and find the corresponding value. Two points are sufficient to draw a line, but we usuall ask for three. The third point allows us to check the accurac of our answers. Now sketch a graph of this equation, + 8, on ou graph paper. Intercepts Gives the point (, ) The values on the left side of the chart are values for a point. The values on the right side of the chart are the corresponding values. We have alread talked about graphing equations using ordered pairs, but we can also graph linear equations b using the intercept and the intercept. The and intercepts are ordered pairs. The intercept is the point(s) where a graph crosses the ais The intercept is the point(s) where a graph crosses the ais To find the intercept, let be zero and find. To find the intercept, let be zero and find. Eample : Let s look at the equation, + 6. To graph b intercepts: 0 Replace b 0 and find. 0 Replace b 0 and find. When we graph a linear equation using intercepts, we use onl the two points. If the  0, 0, we find another point or two. and intercept is ( )
4 Here are the steps: intercept intercept ( ) ( ) So the intercept is (, 0) and the intercept is ( 0, ). Eample 4: Now ou tr to find the intercepts for the equation 4 1. a. Start b finding the intercept b. Now, find the intercept c. Create a ttable replace b 0 replace b 0 To Graph an Equation To graph equations in two variables: 1) Plot points that make the equation true. Find these points b choosing a value for either or and then solving for the value of the other variable. ) Then draw a line (also called a curve) through the points. It is like the dottodot pictures ou ma have done as a child. We will concentrate on linear equations in this material. However, the process is the same no matter the form of the equation our given. Remember that the graph of ever linear equation in two variables is a line. Onl two points are required to graph a linear equation.
5 Graphing Eamples In these eamples we want to determine whether each equation is linear or not and then graph the equation. Here are some suggestions: Find the  and intercepts, if possible. Plot the intercept and the intercept on the coordinate plane. Draw a line (or curve) through the points. (Pla connectthedots.) Eample 5: 4 Is the equation linear or nonlinear? The equation is linear since both the eponents of and are one. Now find the intercept and intercept. To find the intercept, set 0. ( 0) intercept _(4,0) To find the intercept, set 0 ( ) intercept _(0, 4) Now, ou tr several. Determine whether each equation is linear or not and then graph the equation. Eample 6: 6 5 Is the equation linear or nonlinear? Wh? Now, find the intercept and intercept. intercept intercept
6 Eample 7: Is the equation linear or nonlinear? Wh? Now, find the intercept and intercept. intercept intercept If the and intercepts are the same, find two more points. Select values for and find or select values for and find. Eample 8: Is the equation linear or nonlinear? Wh? Now, find the intercept and intercept. intercept intercept
7 Eample 9: Here is an interesting problem. On the same set of aes, graph,, and,. Create a table for each equation, to organize our points. (This activit is a good place to use those map colors ou never used in high school.) Describe the differences and similarities in these graphs. Eample 10: And here is another interesting problem. On the same set of aes, graph, +, and 4. Describe the differences and similarities in these graphs. In summar, a linear equation in two variables looks like + A B C in standard form. An equation is onl linear if the variables have eponents of one. We can graph linear equations b plotting points or using the intercept and intercept. Remember that we onl need two points to graph the equation of a line. If the intercept and intercept are the same, then we need to find additional ordered pairs to help us graph the linear equation.
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