Lesson eleven- seventeen: More Proofs

We now turn our attention to evaluating and proving expressions. Imagine that someone asks you to prove that x = x, or that 1 + 2 = 2 + 1. How would you prove such a simple thing? In this section, we will work with axioms to prove equations.

Here is an important list of axioms that you will need to use while proving equations:

Law 1:

To see the process of proving equations, let’s work through the following example:

Example 1: Prove x = x.
Step 1: Write the given example → x = x
Step 2: Use law 1 → a + b = b + a
Step 3: Replace b + a with a + b → a + b = a + b
Step 4: Replace b with 0 → a + 0 = a + 0.
Remember that the command affects both sides of the equation equally! Step 5: Replace a + 0 with a → a = a
Step 6: Replace a with x → x = x.

Here are the very important steps for evaluating an expression. To “evaluate an expression” means to follow a number of steps that lead to the result:

  1. Write the given expression
  2. Review the axioms and use one that is most similar in operation and number
  3. Rearrange the axiom if necessary to more closely resemble the equation
  4. Continue to use laws to replace values
  5. Substitute
  6. Do the first operation that applies
    1. Add
    2. Subtract
    3. Multiply
    4. Divide
  7. Keep doing this until there is nothing to do.
  8. Underline the answer.

Example 2: Prove x + 1 = 1 + x. This time you have to write on the line.
Step 1: Write the given equation → ______________________
Step 2: Choose an axiom that is similar in operation and number → _________________
Step 3: Replace a with ____ → _______________________
Step 4: Replace b with ____ → _______________________
Step 5: Underline the answer →

Now we are ready to sign on the computer. Open up an Internet browser such as Firefox, Internet Explorer, or Safari, and enter in the address: http://server17.how-why.com




Once you see this page, click on “A proof”.




Now you can begin to prove the equations. To view the axioms, click on the “these axioms” link.




These are the axioms that you will use to prove the equations in this series. Go back to the first page and type in “ Use law 1.” Make sure it is spelled correctly and that you have correct punctuation! Click “Submit it”.

Follow all of your steps for solving a proof until you are finished. Remember, your last entry should be “Underline the answer.”

Once you have finished with the first proof, try the second one.

More practice proving equations
Example 1: Prove 4 • (3 + x) = 12 + 4 • x
Step 1: Write the given equation → _____________________________
Step 2: Use an axiom that is most similar in operation and number → ________________
Step 3: Replace ___________________________



Step : Underline the answer →

Example 2: Prove x + (-x) = y + (-y)
Step 1: Write the given equation → _____________________________
Step 2: Use an axiom that is most similar in operation and number → ________________
Step 3: Replace ___________________________



Step : Underline the answer →

  1. Write the given expression
  2. Review the axioms and use one that is most similar in operation and number
  3. Rearrange the axiom if necessary to more closely resemble the equation
  4. Continue to use laws to replace values
  5. Substitute
  6. Do the first operation that applies
    1. Add
    2. Subtract
    3. Multiply
    4. Divide
  7. Keep doing this until there is nothing to do.
  8. Underline the answer.

Let’s look at a couple of examples.

Evaluate 2 + 3 ÷ x, if x is 2.

Step 1: Write the given expression → 2 + 3 ÷ x
Step 2: Substitute 2 for x → 2 + 3 ÷ 2
Step 3: Divide 3 by 2 → 2 + 1.5
Step 4: Add 2 and 1.5 → 3.5
Step 5: Underline the answer → 3.5

Did you notice how we had to divide first before we could add. Although you do not follow from left to right, division is a higher order than addition, so you must divide before you add. Otherwise the answer would be different.

Evaluate (3+5) • 4

Step 1: Write the given expression → (3 + 5) • 4
Step 2: Add 3 and 5 → 8 • 4
Step 3: Multiply 8 and 4 → 32
Step 4: Underline the answer → 8

Did you notice how we first had to get rid of the parenthesis around the 3+5? This is because you cannot perform an operation with a parenthesis. You can only perform an operation with a number.

Evaluate 30 – 10 ÷ 2.

Step 1: Write the given expression → 30 – 10 ÷ 2
Step 2: Divide 10 by 2 → 30 – 5
Step 3: Subtract 5 from 30 → 25
Step 4: Underline the answer → Step 1: Write the given expression → 30 – 10 ÷ 2
Step 2: Divide 10 by 2 → 30 – 5
Step 3: Subtract 5 from 30 → 25
Step 4: Underline the answer → 25