Key Concepts
Determine whether a number is a solution to an equation.
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- 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.
- Locate the “equals” word(s). Translate to an equal sign.
- Translate the words to the left of the “equals” word(s) into an algebraic expression.
- Translate the words to the right of the “equals” word(s) into an algebraic expression.
Problem-solving strategy
- Read the problem. Make sure you understand all the words and ideas.
- Identify what you are looking for.
- Name what you are looking for. Choose a variable to represent that quantity.
- 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.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- 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
- Choose one side to be the variable side, and then the other will be the constant side.
- Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
- Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
- Make the coefficient of the variable [latex]1[/latex], using the Multiplication or Division Property of Equality.
- 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
- Find the least common denominator of all the fractions in the equation.
- Multiply both sides of the equation by that LCD. This clears the fractions.
- Isolate the variable terms on one side, and the constant terms on the other side.
- Simplify both sides.
- 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
- Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
- Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
- Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
- 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.
- 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:
- One solution. This is when you find the only value of the variable, such as [latex]x = 5[/latex].
- No solution, DNE (does not exist). This is when a false statement appears, like [latex]4 = 7[/latex].
- 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.
