Algebra: Single Variable Equations

This page is designed to help you better understand how to deal with basic equations (equations containing only one variable) and their uses in algebra. Follow any of the links below to go to that section and start understanding equations!

Basic equations (single variable, variable only on one side of the equation, etc.)
Complex equations
Equations consisting of fractions (product, quotient, and power)
Quadratic equations
Quiz on Single Variable Equations

Basic Equations

Basic equations (equations containing only one variable, etc.) are usually covered in pre-algebra courses. We've done that on this site as well, so if you want to learn about equations that only deal with whole numbers and one variable, follow this link.

Complex Equations

This section will get you on your way to understanding how to deal with equations that contain variables on both sides of the equals (=) sign, equations that contain fractions and/or decimals, and multiple operations.

Important Things to Remember

Every operation you perform on one side of an equation must be performed on both sides of the equals sign.
Add like terms. For an explanation of "like terms," follow this link.

The Tutorial
To solve complex equations, the one thing to remember is that you need to get the variable isolated before you can solve the equation. When dealing with fractions and decimals, be very careful with your multiplication and division operations!

Examples:

1. Solve:    3x = 2x + 1
   Solution: 3x - 2x = 1        By subtracting 2x from each sides, the variables
             x = 1              are all on the same side of the equation.  By
                                combining like terms, the variable is isolated,
                                and the equation is simplified and solved.

2. Solve:    .4x = .2(.6x) - 4
   Solution: .4x = .12x - 4     Following the order of operations, multiplication
             .4x - .12x = -4    is done first.  By subtracting .12x from each
             .28x = -4          side of the equation, the variables are all on
             .28x/.28 = -4/.28  the same side.  Combine like terms to simplify.
             x = -100/7         Divide each side by .28 to isolate the
                                variable.  Once the variable is all alone, the
                                answer is found (it is converted to a fraction
                                because it made more sense than the decimal
                                answer of -14.28571429 (a calculator was used
                                to do the conversion).

Equations Consisting Only of Fractions

Throughout your "math experience," you will occasionally see a problem that needs to be solved that is made completely of fractions. While these problems may be intimidating, they are not too hard to solve.

This section will help you understand how to solve this type of equation.

Important Things to Remember

Fractions are division problems.
Know how to find an LCM. Follow this link for an explanation of LCMs.
Order of operations.

Multiplicative property of equality:

               If a = b, then ac = bc when a, b, and c are real numbers.

The Tutorial
These equations, which are also called rational equations, are easy to solve when you eliminate the denominator. The multiplicative property of equality, which tells us you can multiply both sides of an equation by the same thing and the equation will still be correct, is used exclusively here.

Example:

1. Solve:    y   1   y
             - + - = -
             2   4   6
   Solution: 12y   12   12y       The LCM of the denominators is 12.  Multiply
             --- + -- = ---        each numerator by the LCM.  Cancel out the
              2    4     6         denominators to rid the problem of 
                                   fractions.
             6y + 3 = 2y           The denominators are canceled out.
                                   Now, solve for y.
             4y = -3               By subtracting 2y and 3 from each side, the
                                   equation is simplified to something we can
             y = -3/4              easily deal with.
                                   By dividing each side by 4, y is isolated
                                   and the answer is found.

Quadratic Equations

Quadratic equaions are a very complex sort of equation that are easiest to solve by going through a process known as factoring. These equations are second degree polynomial equations. Quadratic equations are so complicated because of the factoring and the fact that they can have 1 or 2 solutions.

This section assumes you know how to factor, and will help you understand quadratic equations.

Important Things to Remember

Quadratic equations are written in the following form: ax^2 + bx + c = d. When d equals zero, the equation is said to be in standard form.

Zero Factor Theorem:

               If pq = 0, then either p or q or both are equal to 0 if p and q are real numbers.

The Tutorial
Keeping the Zero Factor Theorem in mind is the key to solving quadratic equations. For example, if you factor the equation x^2 + 2x - 15 = 0 you will get (x - 3)(x + 5) = 0. By the definition of the Zero Factor Theorem, we know that one or both of those factors has to equal zero.

Example:

1. Solve:    x^2 - x = 42
   Solution: x^2 - x - 42 = 0     Write the equation in standard from.
             (x - 7)(x + 6) = 0  Factor the equation.
             x - 7 = 0           The Zero Factor Theorem says that
             x + 6 = 0           either one factor or both must equal
                                 zero, so we set each factor equal to
                                 zero and solve for x.
             x = 7               Each factor has its own answer.
             x = -6              Since you can only plug one number
             x = 7 or -6         into the the original equation to
                                 see if it works, the answer is 
                                 written with the word or
                                 separating the answers.

Take the quiz on single variable equations. The quiz is very useful for either review or to see if you've really got the topic down.

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