Problem 54 - Introductory Algebra for College Students

Question

Graph equation. \(x+5=0\)

Step-by-Step Solution

Answer

The graph of the equation \(x+5=0\) is a vertical line that passes through the point x=-5.

Step 1: Identify Vertical Line

Recognize that the equation \(x + 5 = 0\) is in the form of a vertical line. This can be deduced because there is no 'y' variable in the equation.

Step 2: Solve for x

Solving for x, subtract 5 from both sides of the equation: \[x = -5\]. Therefore, the line is vertical and crosses the x-axis at x=-5.

Step 3: Graph the Equation

Graph a line parallel to the y-axis that passes through the point x=-5. Since it's a vertical line, all points on this line have an x-coordinate of -5.

Key Concepts

Vertical Line Equation

Understanding vertical lines in the coordinate plane is essential when dealing with linear equations. A vertical line equation is surprisingly simple, it takes the form of \(x = a\), where \(a\) is the x-coordinate through which the line runs. Unlike a typical line with a slope, a vertical line has no slope—or, one might say, an undefined or infinite slope, because it doesn't rise or run in the mathematical sense; it just goes straight up and down.

Let's consider the given exercise equation \(x + 5 = 0\). When you solve this algebraic equation—by subtracting 5 from both sides—you identify that \(x = -5\). This signals that the line is vertical since every point on this line has the same x-coordinate, namely -5. A helpful way to remember this is to think of a vertical flagpole; no matter how high you climb (changing the y-coordinate), you're always at the same spot on the x-axis (which in this case is -5).

Solving Algebraic Equations

To solve algebraic equations, one must understand the process of simplifying and isolating variables. The aim is to find the value(s) of the unknown variable(s) that make the equation true. With linear equations like the one in our exercise \(x + 5 = 0\), the process typically involves basic arithmetic operations—adding, subtracting, multiplying, and dividing—to get the variable by itself.

For this equation, you subtract 5 from each side to isolate the variable 'x' on one side, giving us \(x = -5\). This step is vital in revealing the nature of the graph as a vertical line. Remember that finding the solution to an algebraic equation is like detective work; you're collecting clues (steps) to uncover the hidden value of the variable.

Plotting Points on a Coordinate Plane

Plotting points is the foundation of graphing in mathematics. To plot a point on the coordinate plane, you need an ordered pair, which consists of an x-coordinate and a y-coordinate, written as (x, y). When it comes to a vertical line like in our example, \(x = -5\), every point on this line has an x-coordinate of -5, and the y-coordinate can be any number.

To visualize this, imagine placing a dot at \(x = -5\) for any given y-value. Whether y is 0, 100, or -100, the point will still lie on the line since its x-coordinate remains -5. By plotting several points, such as (-5, 0), (-5, 3), (-5, -3), you will see a straight line forming, which is parallel to the y-axis. This method is a reliable way to graph linear equations and helps provide a clear visual representation of the solution.