Problem 58 - College Algebra Essentials

Question

Graph each equation in a rectangular coordinate system. $$3 x+12=0$$

Step-by-Step Solution

Answer

The Graph of the equation \(3x + 12 = 0\) in a rectangular coordinate system is a straight line parallel to the y-axis passing through the point \((-4,0)\)

Step 1: Solve the Equation

First, arrange the equation by sliding the constant term to the other side to determine the value of \(x\). So, \(3x + 12 = 0\) becomes \(3x = -12\). To find the value of \(x\), divide both sides by the coefficient 3. Doing this results in \(x = -4\).

Step 2: Plotting on the Graph

Now, plot this point on the \((x, y)\) plane. As it's a linear equation, the straight line passing through the point \((-4,0)\) and parallel to the y-axis represents this equation. This is because the equation is independent of \(y\), hence, for every value of \(y\), \(x\) will always be -4.

Key Concepts

Rectangular coordinate system

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane used for graphing equations. It consists of two perpendicular lines, or axes: the horizontal axis known as the x-axis, and the vertical axis known as the y-axis. These axes intersect at a point called the origin, denoted as (0,0). This system helps us place points on a plane using pairs of numbers called coordinates.

Each point is represented by \(x, y\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance. This system allows us to visually represent equations and understand their behavior easily by plotting the points or lines that satisfy them.

Linear equations

Linear equations are equations that form straight lines when graphed on a coordinate system. They have the general form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. These equations show a direct relationship between the variables \(x\) and \(y\).

In many cases, you solve for one variable in terms of the other to draw the graph. For example, solving for \(x\) will tell you how \(x\) changes with \(y\). This is central in analyzing the slope and intercepts of lines on a graph. Linear equations help us quickly identify relationships in equations just by visual inspection, making them advantageous for problem-solving.

Vertical line graph

A vertical line graph results from an equation in which the x-value is constant, as seen in the equation \(3x + 12 = 0\). Solving this gives \(x = -4\), meaning for any \(y\)-value, \(x\) remains \(-4\).

This type of line runs parallel to the y-axis and indicates no relationship between \(x\) and \(y\); hence, there's no slope. You plot it by drawing a vertical line at the constant x-value. It's useful in identifying constraints or fixed conditions within real-world scenarios.

Equation solving steps

Solving equations involves finding the value of the variable that makes the equation true. Let's look at \(3x + 12 = 0\).

• First, rearrange the equation by moving constants to one side, which transforms \(3x + 12 = 0\) into \(3x = -12\).
• Next, isolate the variable by dividing all terms by the constant coefficient, giving \(x = -4\).

This approach simplifies equations, allowing you to easily find and interpret solutions. Understanding each step in solving keeps problems manageable and clear, particularly when graphing or evaluating algebraic expressions.