Algebra Handout, Exercises of Algebra

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Math 063
075

Special types of equations:

Identity, Contradiction (No solution), Conditional

You have practiced solving equations which look like this: 3x + 7 = −2. You would proceed as follows

3 x + 7= − 2 − 7 − 7 3 x = − 9 x = − 3

This type of equation with one (or several) solutions is called a conditional equation. If you substitute the solution back into the original equation and evaluate both sides you will get a true statement, like −2 = −2.

There are 2 other types of equations we will look at now.

1. Solve 5(x + 3) + 2x − 4 = 7x + 11

First we would multiply the bracket by the 5, combine like terms, etc.

5(x + 3) + 2x − 4 = 7x + 11 5 x + 15 + 2x − 4 = 7x + 11 7 x + 11 = 7x + 11

That seems strange to get the identical expression on both sides. Let’s keep going.

7 x + 11 =7x + 11 − 7 x − 7 x 11 =

The x has disappeared from the equation and now we are left with the true statement 11 = 11. When this happens, the original equation 5(x + 3) + 2x − 4 = 7x + 11 is called an identity and the solution is all real numbers. This means you can choose any real number, like 3, 0 or −25, to substitute in for x in the original equation, evaluate both sides and you will get a true statement (like 32 = 32 if you chose x = 3).

If you are asked to solve an equation and it turns out to be an identity, you can state for your answer: The solution is all real numbers. The equation is an identity.

You have seen that conditional equations have one (or several) solutions, and identity equations have all real numbers as their solution. We will now look at a type of equation called contradictory equations (or contradictions) which have no solutions.