Solving Linear Equations

Checking for Solutions to Equations

To determine if a value is a solution to an equation, substitute the value for the variable and simplify both sides. If both sides are equal, the value is a solution.

• Substitution: Replace the variable with the given value.

• Simplification: Perform arithmetic to check equality.

• Example: For , substitute : (Yes, 13 is a solution).

Equivalent Equations

Equations with the same solutions are called equivalent equations. Adding or subtracting the same value from both sides of an equation preserves equivalence.

• Example: and are equivalent.

Objective: Solving an Equation

Solving an equation involves isolating the variable using inverse operations.

1. Get the variable alone.

2. To remove numbers, use the opposite operation (e.g., subtract if added).

3. Check your answer by substitution.

Solving Equations with Fractions and Decimals

Solving Equations with Fractions

When equations contain fractions, multiply both sides by the reciprocal or a common denominator to clear fractions.

• Example: Multiply both sides by :

Solving Equations with Decimals

Multiply both sides by a power of 10 to clear decimals.

• Example: Multiply both sides by 10:

Combining Like Terms and Solving Multi-Step Equations

Combining Like Terms

Combine terms with the same variable to simplify equations. Move variables to one side if they appear on both sides.

• Example:

• Example:

Applications with Percents

Percent as a Ratio

Percents are ratios out of 100. To convert a percent to a decimal, move the decimal two places to the left.

• Example:

• Example:

Solving Percent Problems

Translate percent statements into equations using the formula: is = percent × of.

• Example: What is 25% of 200?

• Example: What percent of 40 is 14?

Word Problems with Percents

Apply percent concepts to real-world scenarios, such as population statistics or discounts.

• Example: If 1,942 million students graduate and this is 65% of all students, total students = million.

• Example: If a price is reduced by 18%, new price = original price × (1 - 0.18).

Evaluating Formulas and Solving for Variables

Evaluating Formulas

Substitute known values into formulas to solve for unknowns. Simplify the expression to find the answer.

• Example: For perimeter , if , ,

Solving for a Specified Variable

To solve for a variable in a formula, isolate it using inverse operations.

• Example: Solve for :

• Example: Solve for :

Geometry Formulas

Common area and perimeter formulas for basic shapes:

• Rectangle:

• Triangle:

• Circle: , Circumference:

• Trapezoid:

Summary Table: Clearing Fractions and Decimals

Equations are easier to solve when fractions or decimals are cleared.

Additional info:

These notes cover foundational concepts in solving linear equations, working with fractions and decimals, percent applications, and formula evaluation, all of which are core topics in Beginning Algebra.