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.
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- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
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- 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.
- Properties of Equalities
Subtraction Property of Equality: For any real numbers [latex]\mathit{\text{a, b,}}[/latex] and [latex]\mathit{\text{c,}}[/latex]
if [latex]a=b[/latex], then [latex]a-c=b-c[/latex].
Addition Property of Equality: For any real numbers [latex]\mathit{\text{a, b,}}[/latex] and [latex]\mathit{\text{c,}}[/latex]
if [latex]a=b[/latex], then [latex]a+c=b+c[/latex].
Division Property of Equality: For any numbers [latex]\mathit{\text{a, b,}}[/latex] and [latex]\mathit{\text{c,}}[/latex] where [latex]\mathit{\text{c}}\ne \mathit{0}[/latex]
if [latex]a=b[/latex], then [latex] \Large\frac{a}{c}= \Large\frac{b}{c}[/latex]
Multiplication Property of Equality: For any real numbers [latex]\mathit{\text{a, b,}}[/latex] and [latex]\mathit{\text{c}}[/latex]
if [latex]a=b[/latex], then [latex]ac=bc[/latex]
- Summary of Fraction Operations
- Fraction multiplication: Multiply the numerators and multiply the denominators.
[latex]\Large\frac{a}{b}\cdot\Large\frac{c}{d}=\Large\frac{ac}{bd}[/latex] - Fraction division: Multiply the first fraction by the reciprocal of the second.
[latex]\Large\frac{a}{b}+\Large\frac{c}{d}=\Large\frac{a}{b}\cdot\Large\frac{d}{c}[/latex] - 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.
[latex]\Large\frac{a}{c}+\Large\frac{b}{c}=\Large\frac{a+b}{c}[/latex] - 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.
[latex]\Large\frac{a}{c}-\Large\frac{b}{c}=\Large\frac{a-b}{c}[/latex]
- Fraction multiplication: Multiply the numerators and multiply the denominators.
- Simplify complex fractions.
- Simplify the numerator.
- Simplify the denominator.
- Divide the numerator by the denominator.
- Simplify if possible.
- Solve equations with fraction coefficients by clearing the fractions.
- 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.
- Solve using the General Strategy for Solving Linear Equations.
- 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.
- 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.