1.37 Linear Equation Graph Definitions
Learn how linear equations are visually represented through graphs, exploring key definitions and foundational concepts in elementary algebra.
Linear Equation Graph Definitions establishes the vocabulary for the visual representation of a linear equation on the coordinate plane, describing the complete set of plotted points that constitute this representation, along with the two special points where that representation crosses each of the coordinate axes.
Linear Equation Graph
A linear equation graph is the set of every point on the coordinate plane whose coordinates satisfy a given linear equation, which, when plotted together, always forms a straight line extending infinitely in both directions. Every point lying on this line represents an ordered pair that makes the linear equation true, and every point not lying on the line represents an ordered pair that makes it false.
Horizontal Intercept
The horizontal intercept, commonly called the x-intercept, is the point at which a linear equation's graph crosses the horizontal axis, occurring where the vertical coordinate equals zero. The horizontal intercept can be found algebraically by substituting 0 for the dependent variable in the equation and solving for the independent variable, and it is often interpreted, in an application context, as the point at which an output quantity first reaches zero.
Vertical Intercept
The vertical intercept, commonly called the y-intercept, is the point at which a linear equation's graph crosses the vertical axis, occurring where the horizontal coordinate equals zero. The vertical intercept can be found algebraically by substituting 0 for the independent variable in the equation and solving for the dependent variable, and in slope-intercept form, y = mx + b, this value is given directly by the constant b without any further calculation needed.
Together, these definitions describe the graph of a linear equation as the full continuous set of coordinate pairs satisfying that equation, and identify the horizontal and vertical intercepts as the two specific points where this graph meets each coordinate axis, both of which can be computed directly from the equation and often carry meaningful interpretations within an applied context.