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A linear equation is a type of algebraic equation that forms a straight line when graphed. The general form is a first-degree equation, meaning the highest power of the variable is one. In this exercise, we start with the equation \(3x + 7y = 21\). This is a classic example of a linear equation because it involves no exponent higher than one for the variables \(x\) and \(y\).
Linear equations are foundational in algebra and appear in various applications, from economic modeling to physics and statistics. When dealing with linear equations:

• Each term in the equation is either a constant or a product of a constant and a single variable.
• The graph will always be a straight line.
• They can often be manipulated into different but equivalent forms, such as the slope-intercept form.

Understanding linear equations is crucial because they provide a beginning point for exploring more complex types of equations and mathematical relationships.

The slope-intercept form is a way to express a linear equation. It is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. Upon transforming our original equation \(3x + 7y = 21\) into the slope-intercept form, we obtained \(y = -\frac{3}{7}x + 3\).

Understanding the components of this form:

• Slope (\(m\)): The slope \(-\frac{3}{7}\) indicates the steepness and direction of the line. A negative slope means the line descends from left to right.
• Y-intercept (\(b\)): The y-intercept \(3\) is the point where the line crosses the y-axis. This is when \(x = 0\).

The slope-intercept form is incredibly helpful because it immediately provides you with important information about the line, making graphing straightforward.

To determine whether an equation defines \(y\) as a function of \(x\), each value of \(x\) must correspond to exactly one value of \(y\). In the exercise, we concluded that \(y\) is a function of \(x\) because the equation is now in the form \(y = mx + b\), where for each \(x\), there is only one \(y\).

In functions:

• Input (\(x\)): Known as the independent variable, where you get a specific output.
• Output (\(y\)): The dependent variable, which changes based on the input \(x\).
• One-to-One Relationship: For a relationship to be a function, each input should map to only one output.

Verifying that an equation describes a function is crucial in mathematical analysis and ensures consistency in problem-solving across various contexts and disciplines.