2.1 Solve Equations Using the Subtraction and Addition Properties of Equality

To Determine Whether a Number is a Solution to an Equation
1. Step 1. Substitute the number in for the variable in the equation.
2. Step 2. Simplify the expressions on both sides of the equation.
3. Step 3.
  Determine whether the resulting statement is true.
  - If it is true, the number is a solution.
  - If it is not true, the number is not a solution.
Addition Property of Equality
- For any numbers a, b, and c, if $a = b$ , then $a + c = b + c$ .
Subtraction Property of Equality
- For any numbers a, b, and c, if $a = b$ , then $a - c = b - c$ .
To Translate a Sentence to an Equation
1. Step 1. Locate the “equals” word(s). Translate to an equal sign (=).
2. Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
3. Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.
To Solve an Application
1. Step 1. Read the problem. Make sure all the words and ideas are understood.
2. Step 2. Identify what we are looking for.
3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem and make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.

2.2 Solve Equations using the Division and Multiplication Properties of Equality

The Division Property of Equality—For any numbers a, b, and c, and $c \neq 0$ , if $a = b$ , then $\frac{a}{c} = \frac{b}{c}$ .
When you divide both sides of an equation by any non-zero number, you still have equality.
The Multiplication Property of Equality—For any numbers a, b, and c, if $a = b$ , then $a c = b c$ .
If you multiply both sides of an equation by the same number, you still have equality.

Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation
1. Step 1. Choose which side will be the “variable” side—the other side will be the “constant” side.
2. Step 2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
3. Step 3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
4. Step 4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
5. Step 5. Check the solution by substituting it into the original equation.

General Strategy for Solving Linear Equations
1. Step 1. Simplify each side of the equation as much as possible.
  Use the Distributive Property to remove any parentheses.
  Combine like terms.
2. Step 2. Collect all the variable terms on one side of the equation.
  Use the Addition or Subtraction Property of Equality.
3. Step 3. Collect all the constant terms on the other side of the equation.
  Use the Addition or Subtraction Property of Equality.
4. Step 4. Make the coefficient of the variable term to equal to 1.
  Use the Multiplication or Division Property of Equality.
  State the solution to the equation.
5. Step 5. Check the solution.
  Substitute the solution into the original equation.

Strategy to Solve an Equation with Fraction Coefficients
1. Step 1. Find the least common denominator of all the fractions in the equation.
2. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.
3. Step 3. Solve using the General Strategy for Solving Linear Equations.

To Solve an Application (with a formula)
1. Step 1. Read the problem. Make sure all the words and ideas are understood.
2. Step 2. Identify what we are looking for.
3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
4. Step 4. Translate into an equation. Write the appropriate formula for the situation. Substitute in the given information.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem and make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.
Distance, Rate and Time
For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula: $d = r t$ where d = distance, r = rate, t = time.
To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all other variables and constants on the other side.

Subtraction Property of Inequality
For any numbers a, b, and c,
if $a < b$ then $a - c < b - c$ and
if $a > b$ then $a - c > b - c .$
Addition Property of Inequality
For any numbers a, b, and c,
if $a < b$ then $a + c < b + c$ and
if $a > b$ then $a + c > b + c .$
Division and Multiplication Properties of Inequality
For any numbers a, b, and c,
if $a < b$ and $c > 0$ , then $\frac{a}{c} < \frac{b}{c}$ and $a c > b c$ .
if $a > b$ and $c > 0$ , then $\frac{a}{c} > \frac{b}{c}$ and $a c > b c$ .
if $a < b$ and $c < 0$ , then $\frac{a}{c} > \frac{b}{c}$ and $a c > b c$ .
if $a > b$ and $c < 0$ , then $\frac{a}{c} < \frac{b}{c}$ and $a c < b c$ .
When we divide or multiply an inequality by a:
- positive number, the inequality stays the same.
- negative number, the inequality reverses.