By: Brooke Norman

 

 

Day 7

Solving Linear Equations by Graphing

 

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Objectives:

1-   To learn how to graphically solve a linear equation

2-   To use the graphing calculator to aid in the learning of how to graphically solve a linear equation

 

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1-   The best way to explain this is by working through examples and explain it as you go.  Keep in mind that the end result is to have an equation in slope-intercept form in order to graph it and find the solution.  Suppose we had the equation of 4x + 6 = 14.  The first step is to subtract 14 from each side in order to get 0 on one side.  You should get 4x-8=0.  The next step is to substitute y in for the 0.  Basically what you are saying is that y equals 0 and they can be substituted for one another.  The new equation should read 4x-8=y.  The student should be able to graph this equation based on the knowledge they learned in the previous lesson.  The y-intercept is -8 and the slope is 4.  Using either pencil and graph paper or a form of graphing software, the graph of this equation should look like the following:

 

         The graph shows every possible x,y value that makes the statement true.  There are indefinitely many points so which one do we pick?  If you go back to the previous step, you will see that we have already set what our y-value must be.  (Remember substituting y for 0?).  When you set y=0 and solve for x, that gives you the x-intercept or where the graph crosses the x-axis.  So, if we solve the equation using y=0, we should get

0=4x-8

4x=8

x=2.

The graph of this equation should cross the x-axis at 2, resulting in the point (2, 0).  We can check the answer by substituting the new x value into the original formula of

4x +6 = 14

4(2) +6 =14

8+6=14

14=14

If the students are still a little confused, here is a basic rundown of it:

         -The original equation given will not have a y variable.

         -Write the equation so that one side is equal to 0.

         -Substitute y in for 0.  The equation should now be in slope-intercept form.  This is able to be graphed.  You also must remember to set the y value equal to 0. 

         -Graph the equation

         -With the equation set equal to 0, find the x-intercept.  This is the solution to the problem and also where it crosses the x-axis.

         -Check your solution in the original equation.

 

Many examples should be worked because there are many steps and concepts the students must understand.  It is very easy for them to get lost along the way.  You may want to go into further details about why the x-intercept is the solution.

 

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2-   This would be a great time to use a graphing calculator.  The example used above was very nice, using nice integers and fractions.  Real life uses will not be so easy; neither will the ones in the text books.  Show the students how to type the equation into their calculator; it must be in slope-intercept form for most graphing calculators.  Then, show them how to use the ÒtraceÓ function to estimate the value of the x-intercept.  An example would be 2.9(5x-6) +11.21-8.4x=x+3.  This is a long and messy equation.  If you move the x+3 to the left side, you have set the equation equal to 0 and may plug it into the calculator to graph its solution. For this example, the answer should be around 1.802, using the graphing calculatorÕs trace function, you find that the x-intercept is between 1.746 and1.9.  Give your students more examples and let the ÒplayÓ around with technology.

 

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Next:  Day 8

Back: Day 6

 

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