Solving Equations

Properties of Equations

Solving equations is a fundamental skill in Intermediate Algebra. Equations are mathematical statements that assert the equality of two expressions. Understanding the properties of equations helps in manipulating and solving them efficiently.

• Addition Property of Equality: If you add the same value to both sides of an equation, the equality remains true.

• Multiplication Property of Equality: If you multiply both sides of an equation by the same nonzero value, the equality remains true.

• Substitution: You can replace an expression with its equivalent value in an equation.

Example:

• Given , subtract 5 from both sides to solve for :

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable has an exponent of one. The general form is .

• Step 1: Simplify both sides of the equation if necessary.

• Step 2: Use the addition or subtraction property to isolate the variable term.

• Step 3: Use the multiplication or division property to solve for the variable.

Example:

• Solve

• Add 3 to both sides:

• Divide both sides by 2:

Applications of Equations

Equations are used to solve real-world problems by translating situations into mathematical statements.

• Word Problems: Identify the unknowns, set up an equation, and solve.

• Example: If a number increased by 8 equals 20, what is the number?

• Let be the number.

• Subtract 8:

Sample Equations and Solutions

• Example 1: (Additional info: This may represent )

• Example 2: Add 20 to both sides:

• Example 3: Add 3:

• Example 4: Subtract from both sides: Add 5: Divide by 2:

Summary Table: Properties of Equality

Additional info: Some equations and examples were inferred from context due to unclear handwriting.