Lesson nineteen – The Inverse Problem

In this lesson, we will explore the inverse problem in which you have the answer given, and you must write the algorithm. It is like working backwards, and it requires a little bit of creativity.

Some review questions: what is the inverse of addition?

What is the inverse of division?

What is the inverse of a fraction?

There are many problems that have a simple inverse, but with an algorithm, the inverse is more complicated.

Sometimes one direction of the process is easy while the other is very complicated.

Example 1

Given x = 5, find a linear equation of this solution.

How would you start? What is always the first step?
Step 1: Write the given answer → x = 5
Step 2: Divide both sides by 2 → x / 2 = 2.5
Step 3: Add 5 to both sides → x / 2 + 5 = 7.5
Step 4: Underline the equation → x / 2 + 5 = 7.5

Example 2

Now find a different equation with the solution x = 5.
Step 1: Write the given answer → x = 5
Step 2: Subtract 3 from both sides → x – 3 = 2
Step 3: Underline the equation → x – 3 = 2

Example 3

Find an equation for x = 5, which is similar to the solution of problem one.

How will we create a similar problem? What is the first step?
Step 1: Write the given solution → x = 5
Step 2: Divide both sides by 3 → x / 3 = 5 / 3
Step 3: Add 4 to both sides → x / 3 + 4 = 5 / 3 + 12 / 3
Step 4: Underline the answer → x / 3 + 4 = 17 / 3

Are these equations really different? The numbers are different, but the operations are the same and there are similar steps.

Let’s compare solving the given equation with find an equation for the given solution.

Solve the given equation for x. Find and equation for the given solution
Step 1 Write the given equation Write the given solutions
Step 2 Do the first instruction that applies
-Do the arithmetic
-Undo the addition
-Undo the subtraction
-Undo the multiplication
-Undo the division
Divide or multiply by a random number, or not
Step 3 Keep on doing this until there is nothing to do Add or subtract a random number, or not
Step 4 Underline the answer Underline the equation

The process of finding an equation for the given solution requires some creativity because it is more open ended. Most people have trouble being creative with this an rely on numbers they are familiar with.