Word Problems:

Here's an easy word problem:

Suzy is ten years older than Billy, and next year she will be twice as old as Billy. How old are they now?

If you don't use algebra, you probably have to solve it by trial and error. This sometimes works fine. Here it is a little slow. And, in other problems, it is next to impossible.

1. Translate into equations

With algebra, the solution is easy. The only problem is to convert the above sentence into equations, because equations are what we need to use algebra. How is this?

S=10+B

S+1=2(B+1)

That is a direct translation of the word problem. I used S to represent Suzy's age (this year), and B for Billy's age. I could use other letters, but these are easier to remember. Do you see how this translation is done? "Suzy is ten years older than Billy" is an equation (S=10+B), but is in words instead of symbols. "Next year she will be twice as old as Billy" is a little more complicated, but is just another equation (S+1=2(B+1)).

The translation process seldom gets much more difficult than the above. But, you may have to weed out extra information. If I had started the word problem with, "Suzy is six inches taller than Billy, and...," you are getting extra info which has nothing to do with the problem. You would have an extra equation (Z=6+L, where Z is Suzy's height, and L is Billy's height). You would find that this equation does not affect the other two equations, at all. You could write down this equation, but you would end up solving the other two equations.

2. Solve the equations

This is just regular algebra (two equations and two unknowns). There are several ways to continue:

1. Solve for one variable (in one equation) and substitute in the other equation.
2. Subtract one equation from another (after changing their form, perhaps), to solve for one variable.
3. Use Matrices.
4. Use Determinants.
5. Graph the equations (not as easy).

If we solved for one variable, then we go back and solve for the other variable. Let's use the first method. S=10+B. Substitute that into the second equation:

10+B+1=2(B+1)

B=9

Going back into the first equation, we find that S=19. Billy is 9, Suzy is 19.