Chapter 3-Equations and Inequalities in Two Variables and Functions

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Linear equations in two variables

an equation that can be written in the for Ax + By = C, where A and B are not both 0.

x-intercept

a point where a graph intercepts the x-axis.

y-intercept

a pint where a graph intercepts the y-axis.

RULE: the y-intercept of y = mx + b

given an equation in the form y = mx + b, the coordinates of the y-intercept are (0, b).

RULE: horizontal lines and vertical lines

the graph of y = c, where c is a real number constant, is a horizontal line parallel to the x-axis with a y-intercept at (0, c).

the graph of x = c, where c is a real number constant, is a vertical line paralell to the y-axis with an x-intercept at (c, 0).

Slope

the ratio of the vertical change between any two points on a line to the horizontal change between those points.

RULES: graph of equations in slope-intercept form

the graph of an equation in the form y = mx + b (slope-intercept form) is a line with slope m and y-intercept (0, b). The following rules indicate how m affects the graph:

If m > 0, then the line slants upward from left to right

If m < 0, then the line slants downward from left to right.

The greater the absolute value of m, the steeper the lined

RULE: the slope formula

given two points (x₁, y₁) and (x₂, y₂), where x₂ ≠ x₁, the slope of the line connecting the two points is given by the formula
m = y₂-y₁/x₂-x₁

RULE: slope of horizontal and vertical lines

two points with different x-coordinates and the same y-coordinates (x₁, c) and (x₂, c) will form a horizontal line with slope 0 and equation y = c.

two points with the same x-coordinates and different y-coordinates (c, y₁) and (c, y₂) will form a vertical line with undefined slope and equation x = c.

RULE: parallel lines

the slopes of parallel lines are equal

RULE: perpendicular lines

the slope of a line perpendicular to a line with a slope of a/b is -b/a.

Linear inequality in two variables

an inequality that can be written in the form Ax + By > C, where the inequality could also be <, ≥ or ≤.

Relation

a set of ordered pairs

Function

a relation in which each value in the domain is assigned to exactly one value in the range

Domain

a set containing initial values of a relation; its input values; first coordinates in ordered pairs.

Range

a set containing all values that are paired to domain values in a relation; its output values; second coordinates in ordered pair.

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