Linear Equations in One Variable

What is a One Variable Linear Equation?

A one variable linear equation contains one letter as a symbol to hold a value; solving an equation provides a solution for the variable requested in the equation. The problem consists of determining what the answer (solution) should be in order to make the equation true. This is the simplest type of equation.

A constant of the equation is a number without a variable attached, for example, −4 or 5. In the equation x - 6 = 10, x is the variable that will be solved for to find a solution, and −6 and 10 are constants.

The solution to this equation, x = 16, solves for the variable x, the placeholder for 16. When we place 16 into the equation to check our work, 16 is substituted for the variable x.

Before we try several example problems, there is an important rule to remember when solving any equation.

Arrows image: Whatever you do to one side of an equation, you do to the other side.

Example 1: How does addition work?

x + 2 = 5

Subtract 2 from both sides

x + 2 - 2 = 5 - 2

x = 3

If we substitute 3 for x, the equation will be true. Check your solution by putting the answer back into the equation. The two sides should be equal when evaluated.

3 + 2 = 5

5 = 5 √

Examining the equation x + 2 = 5, notice that there is three terms in the equation. Terms are separated by a +, - , or =. So x is one term, 2 is another term, and then 5 is the third term.

This is a simple equation that you learned in elementary school, by just learning how to add numbers. Of course there are other equations that might use subtraction, multiplication, and division or a combination of any of these. These are the cases that we will look at in this lesson. Not just the simplest case, but cases that would involve a variety of mathematical concepts.

Here are more examples of one variable linear equations. Notice how the work was formatted above for the equation x + 2 = 5. Notice that the work was done going down the page, keeping the equal, =, sign in the same place. That will help to remember to do the same thing to the other side because the same item will need to be on the other side of the equation as well.

Example 2: How does subtraction work?

x - 3 = 5

Add 3 to both sides

x - 3 + 3 = 5 + 3

x = 8

Check your work by putting the solution (answer) back into the original equation.

8 - 3 = 5

5 = 5 √ equation image indicator

Notice that on the addition and subtraction in Examples 1 and 2, the opposite sign was written under the number to be moved as well as the number. Addition is the opposite of subtraction, and subtraction is the opposite of addition. This is another visual check that you can do, to help verify your math work when you are working down the paper. Your eyes can check for errors quickly if you work down the paper all the concepts are together. When you work across the paper, your eyes will miss something easier and it takes longer to work the problem because you must verify by going back and forth. For many of us our head moves as well and repositioning must take place.

Time and exactness are important to score the best possible on the SAT, so learn to work down the page. Working down the page allows you to "edit" your work quickly and effectively.

Example 3: How does a product (multiplication) work?

=

2x means 2 times x
Divide by 2 on both sides

x=7

Dividing by 2 solves for x
Check your work!

2 * 7 = 14

14 = 14 √

Example 4: How does a quotient (division) work?

= 20

means x divided by 5

5 * = 20 * 5

Multiply 5 to each side to move the 5 to the numerator

x = 100

Multiplying by 5 solves for x
Check your work!

= 20

20=20 √

It is important to realize that these first four examples are the building blocks to use with a variety of combinations of equations. Did you notice that to undo addition, subtraction was used and to undo subtraction, addition was used? The same with multiplication and division, each undid the other.

First

Second

Image Multiplication and division undo each other. Addition and subtraction undo each other.

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