E - Solving Linear Equations Lesson

Solving Linear Equations

Solving equations is one of the most fundamental of all algebraic concepts. In fact, if you think about it, you've been doing algebra since you were in Kindergarten. Remember how your teacher would give you problems like this?

3+ blank = 5

The only difference is now we call that box a variable! See, you really have been doing this for a long time!

We are going to review how to solve linear equations and investigate some possibilities when doing so.

Remember, when solving an equation with one variable, the goal is to get the variable(s) on one side of the equation and the constant terms on the other side. To do this you will use the Properties of Equality for Addition, Subtraction, Multiplication, and Division.

 

x-4=-6 x=-2 Addition Property of Equality: The -4 and the +4 cancel out on the left

5n+3=13 -3. -3. Subtraction Property of Equality 5n=10 /5. /5. Division Property of Equality n=2

 **Remember, when you are solving equations, you can always substitute your answer back in and make sure both sides are, in fact, equal to each other.

4(x+5)=-4 4x+20=-4 Distributive Property of Equality 4x+20-20=-4-20 Subtraction Property of Equality (written horizontally) 4x=-24 /4. /4 Division Property of Equality x=-6

Watch this Khan Academy video for additional practice problems:

In all of these problems, we got an answer for x; one, unique answer. Does that always happen? Let's see...

3(3-x)+5x=2(x+2) 9-3x+5x=2x+4 Distributive Property of Equality 9+2x=2x+4 Combine Like terms 2x-2x+9=2x-2x+4 Subtraction Property of Equality 9-4

WHAT??? 9 is not equal to 4. This is a situation where no matter what we substitute in for x, we will always get an untruth. We say that this problem has no solutions. In other words, no value for x will ever make it true!

Now, let's look at a different scenario.

6-y=5-y+1 6-y+y=5-y+1+y Addition Property of Equality 6=6

This is true, 6 = 6, and our variable cancelled out. This means that ANY value that we use for x will yield a true, balanced equation. We say that this problem has infinitely many solutions, because all real numbers make it true.

In summation, there are three possibilities for the answer to a linear equation.

  1. One solution
  2. No solutions
  3. Infinitely many solutions

Here are some problems for you!

Assignment

Now that you have spent some time learning about solving linear equations, you are ready to complete your Homework. Click here to download the Solving Linear Equations Homework. Links to an external site.

Once you have completed the Solving Linear Equations Homework, check your even answers and make sure you ask your teacher if you have any questions. Links to an external site. When you feel confident in your work, you'll need to take the Solving Linear Equations Homework Quiz. Make sure you have your Homework completed before you log in to take it and remember this is a timed quiz.

IMAGES CREATED BY GAVS