📘intermediate algebra review
3.1 Graph Linear Equations in Two Variables
Last Updated on June 24, 2024
The coordinate plane is a powerful tool for visualizing mathematical relationships. It lets us plot points, graph equations, and analyze functions visually. Understanding how to work with coordinates and graph lines is key to mastering algebra.
Graphing techniques like finding intercepts and using slope make it easier to plot equations accurately. These skills are essential for more advanced math topics and real-world applications. Recognizing special relationships between lines, like parallel or perpendicular, builds a foundation for geometry and calculus.
Graphing in the Coordinate Plane
Points in rectangular coordinates
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- Two-dimensional plane formed by horizontal x-axis and vertical y-axis
- Origin is point of intersection (0, 0)
- Points represented by ordered pair (x, y)
- x-coordinate (abscissa) is horizontal distance from origin
- y-coordinate (ordinate) is vertical distance from origin
- Quadrants determined by signs of x and y coordinates
- Quadrant I (x > 0, y > 0)
- Quadrant II (x < 0, y > 0)
- Quadrant III (x < 0, y < 0)
- Quadrant IV (x > 0, y < 0)
- Points plotted by moving horizontally by x-coordinate and vertically by y-coordinate from origin
Methods for graphing linear equations
- Linear equation in two variables has form
- , , and are real numbers
- and not both zero
- Graph of linear equation is straight line
- Point-plotting method
- Choose two points that satisfy equation
- Plot points on coordinate plane
- Draw straight line connecting points
- Intercept method
- Find x-intercept by setting y = 0 and solving for x
- Find y-intercept by setting x = 0 and solving for y
- Plot intercepts on coordinate plane
- Draw straight line connecting intercepts
- Slope-intercept method
- Rewrite equation in slope-intercept form y = mx + b
- Plot y-intercept (0, b)
- Use slope to find another point
- Draw line through the two points
Vertical vs horizontal lines
- Vertical lines
- Equation in form (a is constant)
- Parallel to y-axis
- Undefined slope
- Horizontal lines
- Equation in form (b is constant)
- Parallel to x-axis
- Slope of zero
Intercepts and Graphing Techniques
Intercepts of linear equations
- x-intercept is point where graph crosses x-axis
- y-coordinate always 0 at x-intercept
- Find by substituting y = 0 and solving for x
- y-intercept is point where graph crosses y-axis
- x-coordinate always 0 at y-intercept
- Find by substituting x = 0 and solving for y
Efficient line graphing techniques
- Slope-intercept form
- is slope, is y-intercept
- Graph by plotting y-intercept and using slope to find another point
- Point-slope form
- is slope, is known point on line
- Graph by plotting known point and using slope to find another point
- Standard form
- , , and are constants
- Find x- and y-intercepts
- Plot intercepts
- Draw line connecting intercepts
Slope and Special Line Relationships
Slope of a line
- Measure of steepness or rate of change
- Calculated using rise over run formula: m = (y₂ - y₁) / (x₂ - x₁)
- Positive slope: line rises from left to right
- Negative slope: line falls from left to right
Parallel and perpendicular lines
- Parallel lines have the same slope
- Perpendicular lines have slopes that are negative reciprocals of each other
Key Terms to Review (28)
Coordinate Plane: The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional graphical representation used to locate and visualize points, lines, and other geometric shapes. It consists of a horizontal x-axis and a vertical y-axis that intersect at a point called the origin, forming a grid-like structure that allows for the precise mapping of coordinates.
Linear Equation: A linear equation is a mathematical expression that represents a straight line on a coordinate plane. It is an equation in which the variables are raised to the first power and the terms are connected by addition or subtraction operations.
Slope-Intercept Form: The slope-intercept form is a way to represent a linear equation in two variables, $y$ and $x$, in the form $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept, the point where the line crosses the $y$-axis. This form provides a straightforward method for graphing linear equations and understanding their key features.
Standard Form: The standard form of an equation is a specific way of writing the equation that provides a clear and organized structure, making it easier to analyze and work with the equation. This term is particularly relevant in the context of linear equations, quadratic equations, and other polynomial functions.
(x, y): (x, y) is a coordinate pair that represents a specific point on a two-dimensional coordinate plane. The x-value corresponds to the horizontal position, while the y-value corresponds to the vertical position of the point. This coordinate pair is fundamental to graphing linear equations in two variables, as it allows for the visual representation of the relationship between the variables.
X-coordinate: The x-coordinate is the horizontal position of a point on a coordinate plane. It represents the distance of a point from the y-axis and is the first value in an ordered pair that describes the location of a point.
Y-coordinate: The y-coordinate is the vertical position of a point on a coordinate plane, measured along the y-axis. It represents the distance of the point from the horizontal x-axis and is used to describe the location of a point in two-dimensional space.
Abscissa: The abscissa is the horizontal axis on a coordinate plane, typically denoted as the x-axis. It represents the independent variable and is used to measure the horizontal position or location of a point on the graph.
Ordinate: The ordinate is the vertical coordinate of a point on a two-dimensional coordinate system. It represents the position of a point along the y-axis, which is the vertical axis perpendicular to the horizontal x-axis.
$ax + by = c$: $ax + by = c$ is the general equation of a linear equation in two variables, $x$ and $y$. This equation represents a straight line in the coordinate plane, where $a$, $b$, and $c$ are real numbers that define the slope and y-intercept of the line.
Point-Plotting Method: The point-plotting method is a technique used to graph linear equations in two variables by identifying and plotting specific points on a coordinate plane. This method involves systematically finding and marking coordinate points that satisfy the given linear equation, which can then be connected to form the complete graph of the line.
Intercept Method: The intercept method is a technique used to graph linear equations in two variables. It involves identifying the x-intercept and y-intercept of the line, which are the points where the line intersects the x-axis and y-axis, respectively. By locating these intercepts, the line can be plotted on a coordinate plane.
Quadrants: Quadrants refer to the four distinct regions created by the intersection of the x-axis and y-axis on a coordinate plane. These four regions are used to organize and analyze the behavior of linear equations and their corresponding graphs.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation crosses the x-axis. It represents the value of x when the function's y-value is zero, indicating the horizontal location where the graph intersects the horizontal axis.
Y-intercept: The y-intercept is the point where a line or graph intersects the y-axis, representing the value of the function when the independent variable (x) is equal to zero. It is a crucial concept in understanding the behavior and characteristics of various types of functions and their graphical representations.
$y = mx + b$: The equation $y = mx + b$ represents a linear function, where $y$ is the dependent variable, $x$ is the independent variable, $m$ is the slope, and $b$ is the $y$-intercept. This equation describes a straight line on a coordinate plane.
Ordered Pair: An ordered pair is a set of two numbers or coordinates that represent a specific point on a coordinate plane. The ordered pair is typically denoted as (x, y), where x represents the horizontal position and y represents the vertical position of the point.
Point-Slope Form: The point-slope form is a way to express the equation of a line given the slope of the line and a single point on the line. It allows you to easily write the equation of a line without having to find the y-intercept first.
$y - y_1 = m(x - x_1)$: $y - y_1 = m(x - x_1)$ is an equation that represents the slope-intercept form of a linear equation in two variables, $x$ and $y$. It describes the relationship between the coordinates of a point $(x_1, y_1)$ on the line and the slope $m$ of the line. This equation is fundamental in graphing linear equations in the Cartesian coordinate system.
Vertical Line: A vertical line is a straight line that runs perpendicular to the x-axis, extending infinitely in the upward and downward directions. It is characterized by a constant x-coordinate, meaning all points on the line share the same x-value.
Slope-Intercept Method: The slope-intercept method is a technique used to graph linear equations in two variables by utilizing the slope and y-intercept of the line. This method provides a straightforward approach to visualizing the relationship between the variables and understanding the behavior of the linear function.
Rise Over Run: Rise over run is a way to express the slope or steepness of a line. It represents the change in the vertical direction (rise) divided by the change in the horizontal direction (run) between two points on a line. This concept is fundamental to graphing linear equations and finding the equation of a line.
X-axis: The x-axis is the horizontal line in a coordinate plane that represents the independent variable or the values along the horizontal dimension. It is the primary reference line for measuring the positions of points along the horizontal direction.
Y-axis: The y-axis is the vertical line on a coordinate plane that represents the vertical or up-and-down dimension. It is used to measure and plot the position of points along the vertical axis of a graph.
Origin: The origin is a fundamental concept in mathematics, particularly in the context of coordinate systems and graphing. It represents the fixed point of reference from which all other points are measured and located on a graph or coordinate plane.
Perpendicular Lines: Perpendicular lines are two lines that intersect at a 90-degree angle, forming a perfect right angle. This geometric relationship has important implications in the context of graphing linear equations, understanding slope, and determining the equation of a line.
Parallel Lines: Parallel lines are a pair of lines that are equidistant from each other and never intersect, maintaining a constant distance between them. This concept is fundamental in the study of linear equations and their graphical representations.
Horizontal line: A horizontal line is a straight line that runs left to right across the plane and has a constant y-coordinate for all points on the line. This means that no matter what value of x is chosen, the y-value will always remain the same. Horizontal lines are essential in graphing because they represent relationships where one variable does not change as the other variable changes.