Problem 53 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$x+1=0$$

Step-by-Step Solution

Answer

The graph for the equation is a vertical line passing through \(x = -1\).

Step 1: Understand the Given Equation

We are given a linear equation \(x + 1 = 0\). We can see that it only contains one variable 'x', which means that 'x' takes a particular value that is independent of 'y'. This implies that our graph will be a vertical line.

Step 2: Solve for x

To get the value of 'x', we will rearrange this equation to solve for 'x'. In doing so, we subtract 1 from both sides of the equation \(x + 1 = 0\), resulting in \(x = -1\). So, the line on our graph will be at \(x = -1\).

Step 3: Graph the Equation

Now that we have the value of 'x', we can graph the equation. It will be a vertical line perpendicular to the x-axis passing through \(x = -1\) on the x-axis. It should be noticed that all the points on this line have x-coordinate -1. The line doesn't depend on the value of 'y', so 'y' is free to take on any value. This vertical line will pass across the entire graph.

Key Concepts

Graphing

Graphing is a visual way to represent equations on a coordinate plane through lines or curves. For each equation, you identify specific points to determine its overall shape and direction. When graphing linear equations like the one given, the first step is to recognize its form. Linear equations can be either horizontal or vertical lines, or they can slant in any direction. In the example of the equation \(x+1=0\), it plots as a vertical line. This is because the equation only concerns the x-variable. Once it’s solved, it provides a fixed x-value causing the vertical line to form, illustrating that x remains constant while y varies. Graphing such equations helps in visualizing mathematical concepts, assisting in understanding where solutions lie on the plane.

Solving Equations

Solving linear equations involves isolating the variable to discover its value, which then determines the line's position on the graph. In the equation \(x + 1 = 0\), we isolate 'x' by performing opposite operations, in this case, subtracting 1 from both sides. This process gives \(x = -1\). Solving equations like this simplifies the task of finding out where the line will be drawn on the x-axis. Understanding how to manipulate these equations is fundamental as it reveals where the graphical representation intersects with the coordinate plane, directly showcasing the relationship between variables involved.

Vertical Lines

Vertical lines are unique as they only focus on the x-coordinates. In a vertical line such as the one from the equation \(x = -1\), every point on this line has an x-value of -1, irrespective of the y-value. This means it stretches infinitely in both directions along the y-axis. These lines stand perpendicular to the x-axis and remain parallel to the y-axis. Vertical lines can be thought of as barriers, signifying that any form of horizontal movement along the x-axis is restricted to the particular x-value. Their representation on graphs serves to clearly denote how an equation's solution affects only one dimension of the coordinate plane.