Problem 23 - Elementary Algebra

Question

Graph the equations. $$ x+1=0 $$

Step-by-Step Solution

Answer

Answer: The equation \(x+1=0\) represents a vertical line that intersects the x-axis at x = -1.

Step 1: Solve the equation for x

To find the x-value that satisfies the equation, we need to solve for x: $$ x+1=0 $$ Subtract 1 from both sides of the equation to isolate x: $$ x=-1 $$

Step 2: Graph the equation

Now that we know x=-1, we can graph the equation. Since there is no y variable included in the equation, this line will be vertical, crossing the x-axis at x = -1. To draw the graph, place a point at (-1, 0) and draw a vertical line through this point extending infinitely in both the positive and negative y directions. This vertical line represents the graph of the equation \(x+1=0\).

Key Concepts

Solving Equations

When we talk about solving equations, we refer to the process of finding the values that make the equation true. For simple linear equations with one variable, the objective is to isolate the variable on one side of the equation. In our case, the equation is \( x+1=0 \). To solve it, we need to perform operations that reverse the ones in the equation until the variable stands alone.

Here's a quick guide to solving such an equation:

• Identify the operations that are being performed on the variable.
• In this instance, the variable \( x \) is being incremented by 1.
• To isolate \( x \), we do the opposite operation on both sides of the equation, meaning we subtract 1 from both sides resulting in \( x=-1 \).

This process of solving is fundamental as it allows us to find the point at which an equation will intersect an axis, which is the first step we take before graphing.

Vertical Line

A vertical line in coordinate geometry is a line that runs up and down the plane, parallel to the y-axis. Unlike horizontal lines, vertical lines are defined by having a constant x-value for all points on the line. This is why vertical lines cannot be represented by the typical \(y=mx+b\) format used for other linear equations, where \(m\) is the slope and \(b\) is the y-intercept. Since the slope of a vertical line is undefined, we use the equation \(x=c\), where \(c\) is the constant x-value.

When graphing \( x+1=0 \), we know that for every value of y, the value of x must remain -1. To visualize this graphically, imagine moving straight up or down from the initial point \( (-1, 0) \) while ensuring that the line remains perfectly straight and never shifts left or right. This creates a vertical line that forever passes through all points with an x-value of -1.

X-intercepts

The x-intercepts of a graph are points where the graph crosses the x-axis. On a coordinate plane, the x-axis is the horizontal line that runs from left to right. To find an x-intercept, we typically set the y-value to zero and solve for x. The x-intercepts are especially useful when graphing linear equations since they signify where the relationship expressed by the equation has no y-component, i.e., where the output of the equation is zero.

Returning to our example, once we have solved the equation \( x+1=0 \) and found that \( x=-1 \), we identified that the graph would intersect the x-axis at \( (-1, 0) \). This is the sole x-intercept of the vertical line because vertical lines, defying the conventionally infinite possibility of x-intercepts for non-vertical lines, cross the x-axis at only one point. Remember, not all linear equations will have x-intercepts, particularly those representing vertical lines where y is not a factor in the equation at all.