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The rectangular coordinate system, also known as the Cartesian coordinate system, is essential for graphing linear equations. It consists of two perpendicular number lines that intersect at the origin, which is the point (0,0). The vertical line is called the y-axis, and the horizontal line is the x-axis. Every point in the plane is defined by a pair of numerical coordinates: the x-coordinate tells us the horizontal position, and the y-coordinate tells us the vertical position.

• The x-axis runs left-right and helps us determine how far a point is from the vertical, y-axis.
• The y-axis runs up-down and helps us determine how far a point is from the horizontal, x-axis.

When graphing an equation like \(3x+12=0\), we use this system to locate the exact position where the equation is true. We plot points that satisfy the equation and then draw the graph accordingly. The ability to navigate this system is foundational for visualizing and solving many types of algebraic problems.

Solving linear equations is the process of finding the value(s) of the variable(s) that make the equation true. A linear equation can have one or more variables but, regardless of the number, the steps to solve them are fairly straightforward.
For example, the given equation \(3x+12=0\) is a linear equation with one variable, x. To solve for x:

• We first isolate the variable on one side of the equation. In this case, we would subtract 12 from both sides to get \(3x = -12\).
• Next, we divide both sides by the coefficient of x, which is 3, yielding the solution \(x = -4\).

Understanding how to manipulate and solve these equations is central to mastering algebra. It enables students to find solutions that are often essential for subsequent steps in more complex problems or to graph the equation accurately on a coordinate system.

Plotting vertical lines on a rectangular coordinate system represents all the points where the x-coordinate is constant. In other words, no matter what value the y-coordinate takes, the x-coordinate remains the same.

Finding the x-intercept

For the linear equation we are dealing with, \(3x+12=0\), after solving for x, we found that \(x = -4\). This tells us our line is vertical since the x-value does not change.

Plotting the line

To plot the line, we:

• Draw a straight vertical line through the x-coordinate of -4, which is our x-intercept.
• Understand that this line will cross through every possible y-coordinate, so any point with an x-coordinate of -4 lies on this line.

Identifying and plotting vertical lines is straightforward and primarily relies on recognizing that the line has a fixed x-coordinate. It's a unique case within linear equations, and an important concept to grasp when studying algebra and geometry.