Key Concepts

• How to determine whether a number is a solution to an equation.
• Step 1. Substitute the number for the variable in the equation.
• Step 2. Simplify the expressions on both sides of the equation.
• Step 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.
• 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.
• Properties of Equalities
 Subtraction Property of Equality: For any real numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$ if $a=b$, then $a-c=b-c$. Addition Property of Equality: For any real numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$ if $a=b$, then $a+c=b+c$. Division Property of Equality: For any numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$ where $\mathit{\text{c}}\ne \mathit{0}$ if $a=b$, then $\Large\frac{a}{c}= \Large\frac{b}{c}$ Multiplication Property of Equality: For any real numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c}}$ if $a=b$, then $ac=bc$
• Summary of Fraction Operations
• Fraction multiplication: Multiply the numerators and multiply the denominators.
$\Large\frac{a}{b}\cdot\Large\frac{c}{d}=\Large\frac{ac}{bd}$
• Fraction division: Multiply the first fraction by the reciprocal of the second.
$\Large\frac{a}{b}+\Large\frac{c}{d}=\Large\frac{a}{b}\cdot\Large\frac{d}{c}$
• Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
$\Large\frac{a}{c}+\Large\frac{b}{c}=\Large\frac{a+b}{c}$
• Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
$\Large\frac{a}{c}-\Large\frac{b}{c}=\Large\frac{a-b}{c}$
• Simplify complex fractions.
1. Simplify the numerator.
2. Simplify the denominator.
3. Divide the numerator by the denominator.
4. Simplify if possible.
• Solve equations with fraction coefficients by clearing the fractions.
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. Solve using the General Strategy for Solving Linear Equations.
• 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$1$, using the Multiplication or Division Property of Equality.
5. Check the solution by substituting into the original equation.
• 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 $1$. 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.