Key Concepts

Determine whether a number is a solution to an equation.

1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
3. Determine whether the resulting equation is true.

If it is true, the number is a solution.
If it is not true, the number is not a solution.

Subtraction and Addition Properties of Equality

• Subtraction Property of Equality

For all real numbers a, b, and c,
if a = b then [latex]a-c=b-c[/latex] .

• Addition Property of Equality

For all real numbers a, b, and c,
if a = b then [latex]a+c=b+c[/latex] .

Translate a word sentence to an algebraic equation.

1. Locate the “equals” word(s). Translate to an equal sign.
2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
3. Translate the words to the right of the “equals” word(s) into an algebraic expression.

Problem-solving strategy

1. Read the problem. Make sure you understand all the words and ideas.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

Division and Multiplication Properties of Equality

• Division Property of Equality:
  • For all real numbers a, b, c, and [latex]c\ne 0[/latex] , if [latex]a=b[/latex] , then [latex]ac=bc[/latex] .
• Multiplication Property of Equality:
  • For all real numbers a, b, c, if [latex]a=b[/latex] , then [latex]ac=bc[/latex] .

Solve an equation with variables and constants on both sides

1. Choose one side to be the variable side, and then the other will be the constant side.
2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
4. Make the coefficient of the variable [latex]1[/latex], using the Multiplication or Division Property of Equality.
5. Check the solution by substituting into the original equation.

The Distributive Property of Multiplication

For all real numbers [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex], [latex]a(b+c)=ab+ac[/latex].

Solve equations by clearing the Denominators

1. Find the least common denominator of all the fractions in the equation.
2. Multiply both sides of the equation by that LCD. This clears the fractions.
3. Isolate the variable terms on one side, and the constant terms on the other side.
4. Simplify both sides.
5. Use the multiplication or division property to make the coefficient on the variable equal to [latex]1[/latex].

Solving Equations of the Form [latex]|x|=a[/latex]

For any positive number [latex]a[/latex], the solution of [latex]\left|x\right|=a[/latex] is [latex]x=a[/latex]  or  [latex]x=−a[/latex].  [latex]x[/latex] can be a single variable or any algebraic expression.

General strategy for solving linear equations

1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
4. Make the coefficient of the variable term equal to [latex]1[/latex]. Use the Multiplication or Division Property of Equality. State the solution to the equation.
5. Check the solution. Substitute the solution into the original equation, to make sure the result is a true statement.

Solutions to equations can fall into three categories:

1. One solution. This is when you find the only value of the variable, such as [latex]x = 5[/latex].
2. No solution, DNE (does not exist). This is when a false statement appears, like [latex]4 = 7[/latex].
3. Many solutions, also called infinitely many solutions or All Real Numbers. This is when a true statement appears, like [latex]x + 3 = x + 3[/latex].

Glossary

solution of an equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

isolate a variable

To isolate a variable means to rewrite an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.