5.5 Graphing Linear Equations and Inequalities
3 min read•june 18, 2024
The coordinate plane is like a map for math. It uses two number lines to pinpoint locations with (x, y) coordinates. This system divides space into four , each with its own sign rules for x and y values.
Linear equations and inequalities create straight lines or shaded regions on this map. By plotting points and connecting them, we can visualize these mathematical relationships. Understanding slopes, intercepts, and half-planes helps us navigate this mathematical landscape.
Graphing in the Coordinate Plane
Plotting points in coordinate systems
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- () forms a two-dimensional plane by intersecting a horizontal number line () and a vertical number line ()
- is the point where the x-axis and y-axis intersect with coordinates (0, 0)
- Points on the coordinate plane represented by (x, y) where x is the horizontal coordinate and y is the vertical coordinate
- Coordinate plane divided into four quadrants by the x-axis and y-axis
- Quadrant I: x > 0 and y > 0 (upper right)
- Quadrant II: x < 0 and y > 0 (upper left)
- Quadrant III: x < 0 and y < 0 (lower left)
- Quadrant IV: x > 0 and y < 0 (lower right)
Graphing Linear Equations and Inequalities
Graphing linear equations
- in two variables, x and y, written in the form where a, b, and c are real numbers and a and b are not both zero
- Graphing a linear equation:
- Find at least two solutions (x, y) that satisfy the equation
- Choose values for x and solve for y, or choose values for y and solve for x
- Plot the solutions as points on the coordinate plane
- Connect the points with a straight line
- Find at least two solutions (x, y) that satisfy the equation
- Graph of a linear equation is a straight line extending infinitely in both directions
- The of the line represents the rate of change between x and y coordinates
- The is the point where the line crosses the y-axis (x = 0)
- The is the point where the line crosses the x-axis (y = 0)
Linear inequalities on coordinate planes
- in two variables, x and y, written in one of four forms:
- Graphing a linear inequality:
- Graph the related linear equation as a
- If the inequality is strict (< or >), the dashed line is not included in the solution
- If the inequality is inclusive (≤ or ≥), the dashed line is included in the solution, so it should be a
- Shade the that satisfies the inequality
- If y is isolated, shade above the line for > or ≥, and below the line for < or ≤
- If y is not isolated, test a point not on the line to determine which half-plane to shade
- Graph the related linear equation as a
- Graph of a linear inequality is a shaded half-plane extending infinitely in all directions, bounded by the line representing the related equality
Relationships between lines
- have the same slope but different y-intercepts
- have slopes that are negative reciprocals of each other
Key Terms to Review (24)
$ax + by \geq c$: The expression $ax + by \geq c$ represents a linear inequality where $a$ and $b$ are coefficients, $x$ and $y$ are variables, and $c$ is a constant. This inequality shows a relationship where the combined weighted values of $x$ and $y$ must be greater than or equal to a certain threshold, $c$. Graphing this inequality involves determining a region on the coordinate plane that satisfies this condition, which is typically represented by a shaded area and a boundary line.
$ax + by \leq c$: $ax + by \leq c$ is a linear inequality that represents a region on a graph, where $a$ and $b$ are coefficients, $x$ and $y$ are variables, and $c$ is a constant. This inequality describes all the points $(x, y)$ that satisfy the relationship defined by the equation $ax + by = c$, including those that lie below the line represented by this equation. Understanding this concept is essential for graphing linear inequalities, as it allows us to visualize the solution set that meets the given criteria.
$ax + by < c$: The expression $ax + by < c$ represents a linear inequality in two variables, where $a$ and $b$ are coefficients, $x$ and $y$ are the variables, and $c$ is a constant. This inequality describes a region on a graph where all the points $(x, y)$ satisfy the condition that the linear combination of the variables is less than the constant. The graph of this inequality consists of a boundary line represented by the equation $ax + by = c$, with the area below this line indicating the solutions to the inequality.
$ax + by = c$: $ax + by = c$ is the standard form of a linear equation in two variables, where $a$, $b$, and $c$ are constants. This format represents a straight line on a coordinate plane, with $a$ and $b$ determining the slope and intercepts of the line. Understanding this equation is crucial for graphing linear equations, as it provides a straightforward way to visualize relationships between two quantities and identify solutions that satisfy the equation.
$ax + by > c$: The expression $ax + by > c$ represents a linear inequality in two variables, where $a$, $b$, and $c$ are constants. This inequality indicates that the values of $x$ and $y$ will create a region on a graph, specifically the area above the line represented by the equation $ax + by = c$. Understanding this term is crucial for identifying feasible regions and solutions in systems of inequalities.
Cartesian coordinate system: The Cartesian coordinate system is a two-dimensional system used to define the position of points on a plane using an ordered pair of numbers, typically represented as (x, y). Each point corresponds to a unique combination of horizontal (x-axis) and vertical (y-axis) distances from a defined origin. This system allows for the easy visualization and representation of linear equations and geometric concepts, facilitating the understanding of relationships between different mathematical objects.
Dashed line: A dashed line is a visual representation used in graphing to indicate a boundary that is not included in the solution set of an inequality. It often appears when graphing linear inequalities, signaling that points on the line do not satisfy the inequality condition. This distinction is crucial for understanding the nature of the solutions represented in the graph.
Half-plane: A half-plane is a geometric concept that refers to one of the two regions formed when a line divides a two-dimensional plane. It is essential for understanding linear equations and inequalities, as it represents the solutions to those equations or inequalities. In particular, when dealing with linear inequalities, the half-plane can indicate all the possible values that satisfy the inequality, extending infinitely in one direction.
Linear equation: A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane, typically expressed in the form $$y = mx + b$$, where $$m$$ represents the slope and $$b$$ represents the y-intercept. This equation shows a direct relationship between two variables, indicating how one variable changes with respect to another. Linear equations can be manipulated and solved for unknowns, making them essential in various applications, from real-world problems to graphical representations.
Linear Inequality: A linear inequality is a mathematical expression that shows the relationship between two values where one value is not equal to the other, typically expressed in the form of $ax + b < c$, $ax + b \leq c$, $ax + b > c$, or $ax + b \geq c$. These inequalities can represent ranges of values rather than single points, and they are used to model situations where constraints exist, making them essential in understanding how to evaluate and compare different scenarios.
Ordered Pairs: An ordered pair is a pair of numbers used to represent a point in a coordinate system, typically written in the form (x, y). The first number, x, indicates the position along the horizontal axis, while the second number, y, indicates the position along the vertical axis. This concept is crucial for understanding how to plot points on a graph and analyze relationships between variables in equations and functions.
Origin: In mathematics, the origin refers to the point of intersection of the coordinate axes in a Cartesian plane, typically represented as (0, 0). This point serves as the reference for all other points in the plane and is critical for defining and understanding linear equations and geometric concepts, acting as the starting point for graphing and analysis.
Origination date: The origination date is the starting date when a loan agreement is officially executed. It marks the beginning of the loan term and the accrual of interest.
Parallel lines: Parallel lines are lines in a plane that never meet or intersect, no matter how far they are extended. They maintain a constant distance apart and have the same slope when represented in a coordinate system, which is essential in understanding relationships between linear equations and geometric properties.
Perpendicular lines: Perpendicular lines are two lines that intersect at a right angle (90 degrees). They are a fundamental concept in geometry and are essential for defining orthogonality in various geometric contexts.
Perpendicular Lines: Perpendicular lines are two lines that intersect at a right angle, forming an angle of 90 degrees. This relationship between lines is essential in various mathematical contexts, especially in geometry and graphing linear equations, as it helps to define slopes and relationships between different lines in a coordinate plane.
Quadrants: Quadrants are the four sections of a Cartesian coordinate system created by the intersection of the x-axis and y-axis. Each quadrant is designated by a number (I, II, III, IV) and contains specific combinations of positive and negative values for x and y coordinates. Understanding quadrants is essential for graphing linear equations, functions, and systems of inequalities since they determine where points lie in the coordinate plane.
Rectangular coordinate system: A rectangular coordinate system, also known as a Cartesian coordinate system, is a two-dimensional plane defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane can be identified by an ordered pair of numbers, (x, y), which represent its horizontal and vertical positions relative to the origin, where the two axes intersect. This system is essential for graphing linear equations and inequalities, as it provides a structured way to visualize relationships between variables.
Slope: Slope is a measure of the steepness or incline of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on that line. It plays a crucial role in understanding relationships in equations and inequalities, helping to determine whether they increase or decrease, and is essential for graphing functions and analyzing systems of equations.
Solid line: A solid line is a type of line used in graphing that represents all the points that satisfy an equation or inequality without any breaks or interruptions. In the context of linear equations and inequalities, a solid line indicates that the points on the line are included in the solution set, which is crucial for understanding the relationships between variables.
X-axis: The x-axis is the horizontal line in a two-dimensional coordinate system that represents the independent variable in a graph. It serves as a reference line from which the position of points is measured, usually indicating the values of the first variable in ordered pairs. Understanding the x-axis is crucial for interpreting and graphing linear equations, inequalities, and relationships between variables.
X-intercept: The x-intercept is the point on a graph where a function or relation crosses the x-axis, meaning that at this point, the value of y is zero. It is a critical concept in understanding linear equations, quadratic equations, and various types of functions, as it provides valuable information about their behavior and characteristics. The x-intercept can be found by setting the output (y-value) of an equation to zero and solving for the input (x-value).
Y-axis: The y-axis is a vertical line in a two-dimensional Cartesian coordinate system that represents the dependent variable in a graph. It is perpendicular to the x-axis, which represents the independent variable, and both axes intersect at the origin, (0,0). Understanding the y-axis is crucial for interpreting relationships between variables and visualizing data effectively.
Y-intercept: The y-intercept is the point where a graph intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. This key feature helps to understand linear relationships, curves, and data trends, providing crucial information for graphing and analyzing equations across various mathematical contexts.