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Linear equations are a core component of understanding algebra and function graphing. A linear equation is one that forms a straight line when graphed on a coordinate plane. The standard form of a linear equation is usually given as \(Ax + By = C\). In this form, \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables. When transforming from parametric to linear form, as seen in the original exercise, it's essential to express both \(x\) and \(y\) in terms of a single parameter \(t\), and then eliminate \(t\) to form a direct relationship between \(x\) and \(y\). This process reveals the linear relationship in its familiar algebraic form. For example, the equation \(y = -1.5x + 4\) represents a linear relationship where \(m = -1.5\) is the slope and \(c = 4\) is the y-intercept.

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry by using coordinates to define and describe geometric figures like lines. In coordinate geometry, you can represent lines using equations, such as linear or parametric equations. It helps us understand the position of points and figures in the plane using ordered pairs (\((x, y)\)).

With the parametric equations given in the problem, you can plot points on the coordinate plane by choosing different values for the parameter \(t\) and calculating corresponding \(x\) and \(y\) values. For instance, at \(t = 0\), you get the point \((2, 1)\), and for \(t = 1\), the point would be \((-1, 3)\). These points lie on the line described by the parametric equations and help visualize the line in the coordinate plane. Understanding this connection between algebraically expressed points and their geometric arrangements is crucial in coordinate geometry.

Graphing lines involves depicting lines on a coordinate grid based on their mathematical equations. With linear equations, you typically start by identifying the slope and the y-intercept from the equation's slope-intercept form, \(y = mx + b\). For the line derived in the exercise, \(y = -1.5x + 4\), the y-intercept (\(b = 4\)) is the point where the line crosses the y-axis. The slope \(m = -1.5\) indicates the steepness and direction of the line.

To graph this line accurately, follow these steps:

• First, plot the y-intercept (0, 4) on the y-axis.
• Next, use the slope \(-1.5\) to determine the next point. From the y-intercept, move one unit right (positive x-direction) and 1.5 units down (due to the negative slope).
• Mark this second point and draw a straight line through both points, extending the line across the grid.

Graphing lines helps visualize solutions and relationships between different variables graphically, making them an important concept in coordinate geometry.