Fundamental Concepts of Algebra

Solving Equations

Solving equations is a foundational skill in algebra, involving finding the value(s) of the variable(s) that make the equation true.

• Linear Equations: Equations of the form can be solved by isolating the variable.

• Quadratic Equations: Equations of the form can be solved by factoring, completing the square, or using the quadratic formula:

• Radical Equations: Equations involving roots, such as , require isolating the radical and then squaring both sides.

• Rational Equations: Equations involving fractions, such as , are solved by finding a common denominator.

Example: Solve :

Equations and Inequalities

Solving and Graphing Linear and Quadratic Equations

Equations can be solved algebraically or graphically. Graphing utilities can approximate solutions to equations that are difficult to solve by hand.

• Graphical Solution: Plot both sides of the equation as separate functions and find their intersection points.

• Algebraic Solution: Manipulate the equation to isolate the variable.

Example: Approximate the solution to using a graphing calculator.

Graphs

Graphing Functions and Analyzing Intercepts and Symmetry

Graphing is a key skill for visualizing solutions and understanding function behavior.

• Intercepts: Points where the graph crosses the axes. The y-intercept is found by setting ; the x-intercept(s) by setting .

• Symmetry: Test for symmetry with respect to the x-axis, y-axis, and origin by substituting , , and into the equation.

Example: For , the y-intercept is (when ).

Functions & Graphs

Distance and Midpoint Formulas

Given two points, the distance and midpoint can be calculated using the following formulas:

• Distance Formula:

• Midpoint Formula:

Example: For points and :

• Distance:

• Midpoint:

Graphs of Circles

Standard Form and Center/Radius

The equation of a circle in standard form is:

• Center:

• Radius:

Example: has center and radius $3$.

Linear Equations and Slope

Slope, Slope-Intercept Form, and Point-Slope Form

The slope of a line measures its steepness and is calculated as:

• Slope-Intercept Form:

• Point-Slope Form:

Example: Find the equation of a line with slope $2(4, 2)$:

Systems of Equations

Solving Linear Systems

Systems of equations can be solved by substitution, elimination, or graphing.

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

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

Example: Solve and by elimination.

Summary Table: Key Formulas and Properties

Additional info:

• This review covers core Precalculus topics: algebraic manipulation, graphing, equations of lines and circles, and basic function analysis.

• Students should be comfortable with both algebraic and graphical approaches to problem solving.