# Deductive Reasoning in Algebra

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After reading this lesson, you'll know how you can solve an algebraic problem by using what you already know is true. This is deductive reasoning. Learn how you can substitute what you know into an equation to help solve it.

## An Algebraic Problem

You and a friend are trying to do some math homework. Your teacher has told you to use deductive reasoning to help you solve these problems. But what is deductive reasoning? Sounds like a fancy term. Actually, all it means is using what you already know to be true. And for algebra, deductive reasoning is an excellent way for you to solve your problems. Take a rule or property that you already know and apply it to the equation that needs to be solved.

Here is the problem you and your friend are currently working on. It's actually a set of three equations. Don't worry. The way this is written allows you to use deductive reasoning to help you solve the problem that much quicker.

## Using Deductive Reasoning

Remember, deductive reasoning is nothing more than using what you already know. So, look carefully at your problem. You need to solve for your variables x, y, and z. Do you see anything that tells you what the values are for x, y and z? Look at the second equation. What does that tell you? It tells you that your x is equal to 1. Aha! You can use that information to help you solve your problem. This is deductive reasoning. You know that x is equal to 1, and you know you can plug in that 1 wherever you see an x.

## Solving the Problem

Let's continue solving the problem. You still can't solve the first equation because you still don't know the values for y or z. You need to eliminate your equation down to one variable in order to solve it. But look at your third equation. It has the variables z and x. True, it has two variables. But don't you already know that x is equal to 1? You do! And you can use it here so you can find your z. You deductively reason that the x here is also equal to 1 and you plug in 1 for x and this is what you get:

• 2z = 1 + 5
• 2z = 6
• 2z / 2 = 6 / 2
• z = 3

You solved for your z by using the properties of algebra that you already know such as dividing both sides by the same number (2) to isolate your variable.

Now, you know both your x and z. That's right! You can now use both of these pieces of information in the first equation to solve that one for y. You deductively reason here that since you already know that your x is equal to 1 and your z is equal to 3 that this is also true for the first equation and you can go ahead and plug these values in to help you solve it.

• 2 (1) + 3y - 3 = 5
• 2 + 3y - 3 = 5
• -1 + 3y = 5
• 3y = 6
• 3y / 3 = 6 / 3
• y = 2

This equation is solved by using known algebra properties such as adding the same number to both sides (-1) and then dividing both sides by the same number (3).

And you are done!

You have found all three of your variables through deductive reasoning. Wasn't that easy?

• x = 1
• y = 2
• z = 3

## Example

Let's look at another example.

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