Solving Equations

In algebra to solve an equation means to find all the numbers that the variable can be replaced with that make the equation true. For example 2 is a solution to the equation x+3=5, because 2+3=5. For a simple equation like that one, we can find the solution by guessing, but as equations get more complicated we need to use fancier methods. One way we can do it is to use a graphing calculator. I have found Pacific Tech's one at  http://www.PacificT.com  quite nice for this, so I will use it, but many other graphing calculators can be used as well.

The way to solve an equation with the Pacific Tech calculator is to put in a separate expression for each side of the equation and click the graph button. The graphing calculator will then draw two curves, and the place where they intersect will be the solution of the equation. To find the exact number you can click on the intersection while holding down the control key and it will show x: something, and that something will be the solution to the equation. There will also be a y: something else, but you can ignore that part.

For example to so it with the above example would look like this.

The purple line represents the left side of the equation, and the red line represents the right side of the equation. What I mean by this is that you can think of the horizontal line as the real number line, and as we travel along it the computer is representing the values of the expressions by the vertical heights. Notice the height of the red line is always 5. This is because the right side of the equation is 5 no matter what x is, but the height of the purple line is different for different values of x. When x is 1, it is 4 reflecting the fact that 1+3=4, when x is 5 it is 8 reflecting the fact that 5+3=8. This is a trial an error method of solving the equation, because in order to draw the two graphs, the computer has to compute x+3 for lots of values of x, but computers are quite fast nowadays, so they can handle this without much trouble.

Of course, we didn't really need the computer to solve this equation, but it will solve many others that will can't so easily guess. Here are some more.

1. 2x-5=17

x=11

2. 2x+3x=-7

x=-1.4

3. 2x+3=5x-9

x=4

4. 3(2x-4)+5(x-7)=5[2(3x-1)-7(x+1)]+-2(2x-5)

x=0.6

5. x²-5x+6=0

x=2,3

6. x(x+1)=6

x=-3, 2

7. x²+x-5=0

x=-2.713, 1.713

8.

x=6

9.

x=15.638

10.

x=1

11.

no solution

12. 2^x=4^x-1

x=2

13. 2^x+1=5^x-1

x=2.5129

14. e^3x+e^-3x=7

x=0.64162, -0.64162

15. log₃x=5

x=243

This one was a little tricky. To enter the log base 3, use the keypad to enter the log base 2, and then change the base. Also I needed to use the zoom button to get the solution on the screen.

16. log₂(x+1)+log₂(x-1)=3

x=3

17. log(logx)=2

It looks like if there is a solution to this one, it is a very large number, so we are going to have to do some serious zooming.

It's getting closer, so perhaps there's some hope.

After lots and lots of mouse clicks it looks like we are making some progress, but we still have a ways to go. This is scientific notation. We are up to 10⁴⁶ now.

And the answer is 10¹⁰⁰. Admittedly this is stretching the capabilities of the graphing calculator. I really was only able to get this with the graphing calculator, because I already knew the answer from analytical methods. This is the sort of problem to make you appreciate analytical methods. You can, however, see it a bit more clearly if you change the number of decimal points in the preferences and zoom in on the y axis.

18. x⁴-3x³-5x²+x-3=0

x=-1.5305, 4.18

19. 2x⁵-3x⁴-2x³+x²-x-7=0

x=2.0979