Algebra Know-It-ALL

262 Two-by-Two Linear Systems

Applying the distributive law, we get

y= (− 4 + 8 π+ 6 − 8 π)/(3− 4 π)

When we add up the terms in the numerator here, we get our reward:

y= 2/(3 − 4 π)

We’ve arrived at our solutions! They are:

x= (− 1 + 2 π)/(3− 4 π)

and

y= 2/(3 − 4 π)

How about some extra credit?

Plug the above numbers into the original equations for x and y, and verify that the answers we got are

correct. You’re on your own! Here’s a hint: (3 − 4 π) divided by itself is equal to 1.

Practice Exercises

This is an open-book quiz. You may (and should) refer to the text as you solve these problems.

Don’t hurry! You’ll find worked-out answers in App. B. The solutions in the appendix may not

represent the only way a problem can be figured out. If you think you can solve a particular

problem in a quicker or better way than you see there, by all means try it!

1. The sum of two numbers is 44. Their difference is 10. What are the two numbers? Use
   the morph-and-mix method.


2. The sum of two numbers is 100. One of them is 6 times the other. What are the two
   numbers? Use the morph-and-mix method.


3. Imagine that you and I are traveling in a car on a level highway at constant speed. I’m
   the driver. There are no other vehicles or living things in sight. You throw a baseball
   straight out in front of the car. The ball strikes the pavement at 135 miles per hour (mi/h).
   Then you throw another baseball directly backward, exactly as hard as the first one. The
   second ball hits the highway, moving opposite to the direction of the car, at 15 mi/h.
   (The ball is not only moving backward relative to the car; it’s also moving backward
   relative to the pavement!) How fast am I driving? How fast do you hurl the baseballs
   relative to the car? Forget about the possible effects of wind and gravity. Here’s a hint:
   Feel free to draw diagrams.


4. The sum of two numbers is −83. Their difference is 13. What are the two numbers?
   Use double elimination.