Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$x+5=0$$

Step-by-Step Solution

Answer

The graph of the equation \(x+5=0\) is a vertical line passing through the point at -5 on the x-axis.

Step 1: Solve the equation for x

To start off, resolve the given equation for x. This involves performing an operation that will isolate x. In this case, subtract 5 from both sides of the equation to find the x value.\n So, \(x+5-5=0-5\), simplifies to \(x=-5\).

Step 2: Graph the solution

Since this is a linear equation in one variable, the graph will be a vertical line at \(x=-5\). Draw a number line, mark the point at -5 and draw a vertical line going through that point.

Key Concepts

Linear equations

Linear equations are a fundamental concept in algebra. They are equations of the first degree, which means that the highest power of the variable is one. A typical linear equation can be written in the form \(ax + b = 0\), where \(a\) and \(b\) are constants and \(x\) is the variable.

Linear equations can describe real-world phenomena such as calculating distances, costs, or anything that requires a constant rate of change. When solving linear equations, the goal is to find the value of \(x\) that makes the equation true. This usually involves isolating the variable by performing arithmetic operations.

In our example, the equation \(x + 5 = 0\) is simple and only needs us to subtract 5 from both sides to solve for \(x\). The solution is \(x = -5\), indicating that a single value satisfies this equation.

Graphing techniques

Graphing linear equations can help us visualize the relationships they depict. For linear equations in one variable like our example, the graph is a straight line.

There are several techniques to graph equations:

• Identify the variable and solve for it.
• For a simpler understanding, visualize the solution as a line on a graph.
• For equations in one variable, such as \(x + 5 = 0\), the line is vertical.

In our case, the equation \(x = -5\) results in a vertical line at \(x = -5\) on the coordinate plane. Unlike the typical \(y = mx + b\) format which would create a sloped line, here you mark the position on the \(x\)-axis and draw a vertical line crossing through it.

Practicing these graphing techniques provides deeper insight into the nature of linear equations.

One-variable equation solutions

Solving linear equations with one variable is straightforward. The process simplifies the equation to find the single solution where it holds true.

For the equation \(x + 5 = 0\), the steps are clear:

• Isolate the variable by performing equivalent mathematical operations on both sides of the equation.
• Here, subtracting 5 from both sides gives \(x = -5\).
• The solution \(x = -5\) can be checked by substituting back into the original equation.

Doing so confirms that the equation holds true, as substituting returns the statement \(0 = 0\).

Understanding one-variable solutions is crucial, as it is a building block for more complex algebraic concepts, such as systems of equations.