BackKey Concepts in Linear Equations and Systems for College Algebra
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Equations & Inequalities
Average Rate of Change
The average rate of change of a function f between two points x_1 and x_2 measures how much the function changes per unit increase in x. It is analogous to the slope of the secant line connecting the points (x_1, f(x_1)) and (x_2, f(x_2)).
Formula:
Interpretation: Represents the change in the function's output divided by the change in input.
Example: If f(x) = 2x, the average rate of change from x = 1 to x = 3 is .
Graphs of Equations
Point-Slope Form of a Line
The point-slope form is used to write the equation of a line when you know a point on the line and its slope. This form is especially useful for constructing linear equations quickly.
Formula:
Variables: (x_0, y_0) is a point on the line, m is the slope.
Example: If a line passes through (2, 3) with slope 4, its equation is .
Slope-Intercept Form of a Line
The slope-intercept form is a standard way to express the equation of a line, highlighting its slope and y-intercept. This form is widely used for graphing and analyzing linear relationships.
Formula:
Variables: m is the slope, b is the y-intercept (the value of y when x = 0).
Example: A line with slope 2 and y-intercept -1 is .
Properties of Parallel and Perpendicular Lines
Understanding the relationship between slopes helps classify lines as parallel or perpendicular.
Parallel Lines: Have equal slopes (m_1 = m_2).
Perpendicular Lines: Have slopes that are negative reciprocals (m_1 = -1/m_2).
Example: If one line has slope 3, a perpendicular line will have slope .
Systems of Equations & Matrices
Definition and Solution of Systems of Equations
A system of equations consists of two or more equations involving the same variables. The solution to the system is the set of values that satisfy all equations simultaneously, often represented as the intersection point(s) of their graphs.
Types: Systems can be linear or nonlinear, but in college algebra, linear systems are most common.
Solution Methods: Graphical, substitution, and elimination methods are used to find solutions.
Example: The system has a solution where the two lines intersect.
