The Intersection of Graphs Method can be used to find approximate (sometimes exact) solutions to equations. In another tutorial, a linear equation was solved by this method. In fact, any type of equation can be solved by this method. Solve the equation x 2 + x = 20 using this method.
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Intersection of Graphs Method of Solving an Equation

This method uses graphing of functions to solve an equation. 

STEP 1: Set Y sub 1 equal to the left side of the equation and Y sub 2 equal to the right side of the equation.

STEP 2: Graph Y sub 1 and Y sub 2 in the same viewing rectangle. 

STEP 3: Locate any points of intersection.  The set of x-values of these points of intersection corresponds to the solution set of the equation. Some solutions may be exact, while others may be only approximate solutions.

 

STEP 1: Press the button. Then enter the left side of the equation for Y sub 1 and the right side of the equation for Y sub 2.

Screen Picture of TI-83 Calculator

STEP 2: Since Y sub 2 has a graph which is a horizontal line at 20 on the y-axis, we will set our initial window as follows: [-10,10,1] by [0,30,2]. This way we are assured that the horizontal line, y = 20, will be visible. After the window is set, press GRAPH. Two intersection points are visible in the screen below.

 

STEP 3:  Now press the 2nd key, then TRACE [Calc], then select 5:intersection. When First curve? appears, press ENTER. When Second curve? appears press ENTER again.

Screen Picture of TI-83 CalculatorScreen Picture of the TI-83 Calculator


Since there are two intersection points, when Guess? appears, move the curser close to one of the points using the ARROW keys and then press ENTER. The screens below show finding the leftmost intersection first.

Screen Picture of TI-83 Calculator

 

The final screen above shows that the intersection point is (-5,20). If there are more points of intersection, move the cursor near each one and repeat the process. The X value of the intersection point, X = -5, is one of the two solutions to the equation. Next repeat the entire process, with the exception at GUESS?, move the curser to the rightmost intersection, then press ENTER. The results are shown below.

The final screen above shows that the intersection point is (4,20). If there are more points of intersection, move the cursor near each one and repeat the process. The X value of the intersection point, X = 4, is one of the two solutions to the equation.

Thus the solutions to the equation, x 2 + x = 20, is x = - 5 or x = 4.

 


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