Linear Equations and Inequalities in One Variable

Evaluating and Solving Linear Equations

Linear equations in one variable are equations that can be written in the form ax + b = c. Solving these equations involves isolating the variable.

• Evaluating Expressions: Substitute the given value for the variable and perform arithmetic operations.

• Solving for a Variable: Use inverse operations to isolate the variable on one side of the equation.

• Example: Solve Subtract 3 from both sides: Divide by 2:

Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.

• Format: where and is an integer.

• Example:

Simplifying Expressions

Simplifying involves combining like terms and applying arithmetic operations.

• Example:

• Order of Operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS)

Linear Equations in Two Variables

Writing Equations of Lines

The equation of a line can be written in slope-intercept form: , where is the slope and is the y-intercept.

• Given Two Points: Use the formula to find the slope.

• Example: Find the equation of the line passing through (1,2) and (3,4).

Graphing Linear Equations

To graph a linear equation, plot the y-intercept and use the slope to find another point.

• Steps:

  1. Plot the y-intercept ().

  2. From the y-intercept, use the slope () to find another point.

  3. Draw a straight line through the points.

Systems of Linear Equations

Solving Systems of Equations

A system of linear equations consists of two or more equations with the same variables. The solution is the point(s) where the equations intersect.

• Methods:

  • Substitution: Solve one equation for one variable and substitute into the other.

  • Elimination: Add or subtract equations to eliminate a variable.

• Example: Solve and .

  1. Solve the first equation for :

  2. Substitute into the second:

  3. Simplify:

  4. Find :

Exponents and Polynomials

Properties of Exponents

Exponents indicate repeated multiplication. Key properties help simplify expressions.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Example:

Polynomials and Factoring

A polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, and multiplication.

• Factoring: Expressing a polynomial as a product of its factors.

• Example: Factor

Rational Expressions

Simplifying Rational Expressions

Rational expressions are fractions with polynomials in the numerator and denominator. Simplification involves factoring and reducing common factors.

• Example:

Solving Equations Involving Rational Expressions

To solve, find a common denominator, clear fractions, and solve the resulting equation.

• Example: Multiply both sides by :

Word Problems and Applications

Translating and Solving Word Problems

Word problems require translating real-world scenarios into algebraic equations.

• Example: "The length of a rectangle is 2 m more than the width. The area is 48 m2. Find the length and width."

  1. Let width = , length =

  2. Area:

  3. Solve:

Additional info:

• Some questions involve graphing and interpreting linear equations, which are foundational for understanding functions and systems.

• Problems cover a range of College Algebra topics, including exponents, polynomials, rational expressions, and applications.