91Ó°ÊÓ

In mathematics, specifically in algebra, the term "two variables" usually refers to an equation involving two unknowns, typically represented as \(x\) and \(y\). These variables can take multiple values, and when they do, they often form a relationship that can be represented graphically.

An equation like \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants, is a common form of a linear equation in two variables. Both \(x\) and \(y\) need to be present to satisfy the condition of being 'two variables'.

When both variables appear in the equation, it creates a situation where for every value of \(x\), there is a corresponding value of \(y\) that satisfies the equation. This is crucial for understanding the graph these equations produce.

• Example: \(3x + 2y = 6\) has two variables \(x\) and \(y\).
• Non-Example: \(y = -1\) only has one variable \(y\) that depends on a constant value.

The "graph of a line" is a visual representation of all the points that satisfy a given linear equation. When dealing with equations in two variables, they typically form a straight line on the Cartesian coordinate system.

Linear equations like \(Ax + By = C\) are represented by straight lines, where each point \((x, y)\) on the line is a solution to the equation. The graph of these equations gives a visual insight into the relationship between \(x\) and \(y\).

Characteristics of the Line Graph:

• Slope: Determines the tilt or steepness of the line, calculated as \(m = -\frac{A}{B}\) for the standard equation.
• Intercepts: Points where the line crosses the axes. The y-intercept is \(C/B\) when \(Beq0\), and the x-intercept is \(C/A\) when \(Aeq0\).

These properties help easily plot and understand the relationship between the variables. However, an equation like \(y = -1\) shows a special graph shape, which we will discuss in the next section.

A "horizontal line" in the context of graphs is a flat line which runs from left to right in a Cartesian plane, and it is characterized by having no slope, or zero slope.

The equation \(y = -1\) is a simple example of a horizontal line equation. Here, the value of \(y\) is constant, meaning for every possible value of \(x\), \(y\) remains \(-1\). This creates a straight line parallel to the x-axis.

Properties of Horizontal Lines:

• Equation Form: Always written in the form \(y = b\), where \(b\) is a constant.
• Slope: The slope is \(0\) because there is no rise over run, or \(\Delta y = 0\).
• Y-Intercept: Passes through the point \((0, b)\).

A horizontal line indicates that the y-value does not change regardless of the x-value, thus it only involves the y variable, and why it cannot be classified as a linear equation with two variables.