Algebra Know-It-ALL

206 First-Degree Equations in One Variable

Are you confused?

Suppose you’re trying to solve a word problem, and you set up an equation in one variable to repre-

sent it. You hope to get a first-degree equation. If you’re lucky, you’ll be able to get the equation into

the form

ax+b= 0

where a and b are constants. However, this might not be possible. If you can’t get the equation into the

standard form for a first-degree equation, you’ll have to use more powerful equation-solving techniques

than those in this chapter. You’ll learn about them in Part 3.

Here’s a challenge!

Try playing the following little number game. This example is only one of infinitely many variations on

the same theme. Choose any real number you want. Perform the following operations in this order:

• Multiply the number you have chosen by 2.


• Subtract 10 from the previous result.


• Multiply the previous result by 2.


• Add 60 to the previous result.


• Divide the previous result by 4.


• Subtract the number you originally picked.


• You will end up with 10, no matter what number you originally chose.

Explain how this trick works.

Solution

Games like this are fabricated by “reverse engineering.” You pick the number that you want the game to

end up at (in this case 10), and then perform multiple operations on successive results to obtain x. You can

see how this example was put together by going through the above sequence of steps backward, replacing

every addition with subtraction, every subtraction with addition, every multiplication with division, and

every division with multiplication. You’ll get this sequence of operations:

• Start with 10.


• Add x,obtaining 10 +x.


• Multiply the previous result by 4, getting 40 + 4 x.


• Subtract 60 from the previous result, getting − 20 + 4 x.


• Divide the previous result by 2, getting − 10 + 2 x.


• Add 10 to the previous result, getting 2x.


• Divide the previous result by 2, getting x.

When you build games of this sort, you must never multiply or divide by any expression containing x. If

you do that, you make it possible for someone to defeat the trick by choosing x such that division by 0

occurs somewhere in the sequence of steps.