# Solving equations using a Ti-84 graphing calculator

You can solve equations (linear, rational, absolute value, trigonometric, logarithmic etc.) and inequalities in a single variable graphically by plotting the left hand side and the right hand side separately and analyzing the intersection of the two graphs.

Here are a few examples:

### Example 1:

What is the value of \(x\) in the equation \(\frac{3}{4}x + 2 = \frac{5}{4}x - 6\)?

a. -16 b. 16 c. -4 d. 4

Step 1: For Y1, enter one side of the equation or inequality Step 2: For Y2, enter the other side of the equation or inequality Step 3: Graph, then adjust window or zoom out so that the intersection, if there, is visible. Step 4: Press 2nd ⇒CALC⇒5 (Intersect) Since there are only two curves, press enter twice and they will be automatically selected. Move the trace close to intersection of curves and then press enter for the guess? Step 5: Read off the value. (This is the value for which the graphs achieve the same height and therefore the left hand side equals the right hand side.) It is your solution. Step 6: Repeat Step 4 and 5 if there are multiple intersections (solutions). |

### Example 2:

What is the solution to the inequality \(\frac{1}{2}x + 3 < 2x - 6\)?

a. x < -5/6 | b. x > -5/6 | c. x < 6 | d. x > 6 |

Procedure is the same as Example 1 except you must note which line is which. [You can get the equation of the curve to appear by selecting trace. Then use up or down arrow to switch between the curves.] We are looking for those values for which the graph of Y1 is below that of Y2. We find the intersection of the two curves, which in this case is at x = 6. Next, we observe that graph of Y1 lies below that of Y2 for x > 6. Therefore, \(\frac{1}{2}x + 3 < 2x - 6\) for \(x>6\) |

### Example 3:

What is the solution set of the equation \(\left| {2x - 1} \right| = 9\)?

a. { } b. {5, -4} c. {-5, 4} d. {5}

** abs ( ) can be found under MATH below the ALPHA key on the calculator

Follow the same steps as example 1.

### Example 4:

In the interval \({0^ \circ } \le \theta \le {360^ \circ }\), how many values of angle satisfy the equation \({\sin ^2}\theta = \frac{1}{4}\)?

a. 1 b. 2 c. 3 d. 4

**Make sure you are in right mode[ Radian or Degree] and choose your graphing window on the calculator accordingly.

Since we are looking in

- choose Xmin close to 0 (below 0 degrees because our interval includes zero degrees and we don't want to miss any solutions)

- choose Xmax to be or slightly larger than 360 degrees.

- Choose Ymax to be 1.5 or just to a comfortable zoom where the intersections are visible.

- Count the intersections of the two curves from 0 to 360 for your solution. The image below shows that one degree value for which sin2(Ø)=1/4 is 150 degrees.

### Questions:

- What does it mean to say that a number is a solution to an equation?
- Is the graph of an equation, a graph of its solutions? For example, is the graph of y = x2 a graph of all points (x,y) that make the left hand side equal the right hand side of the equation when substituted into the equation. Are the points (1, 1) , (2, 4) , (3, 9), (3.14, 3.142 ) solutions? Are they points on the graph?

Now you should have knowledge on how to solve somewhat more complicated equations too. Enjoy!

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