Problem 52 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$x=4$$

Step-by-Step Solution

Answer

The graph of the equation \( x=4 \) is a vertical line passing through the point (4,0) on the x-axis.

Step 1: Understand the Equation

The equation given is \( x=4 \). This equation represents a line in a two-dimensional coordinate system, where all points on the line have an x-coordinate of 4. Since y is not mentioned in the equation, it can take any value. This implies that the line of the graph will be vertical.

Step 2: Plot the Line

To plot the line of the equation \( x=4 \), mark the point (4,0) on the x-axis. From this point, draw a vertical line. This line represents all points where the x-value is 4 and y can be any real number. This creates a vertical line that passes through the point (4,0).

Key Concepts

Coordinate System

When it comes to graphing linear equations, the foundation is the coordinate system. Imagine it as a stage where the mathematical drama unfolds. In its simplest form, the coordinate system is a way to display points on a flat surface using two reference lines, usually labeled the x-axis (horizontal) and y-axis (vertical).

It's essential to grasp that each point in this system has a pair of numbers associated with it, known as coordinates. The first number, or the x-coordinate, tells us how far along the x-axis our point is, while the second number, the y-coordinate, shows how far up or down along the y-axis the point lies.

• An x-coordinate of 3 means move 3 units right from the origin—if it's -3, then we go left instead.
• Similarly, a y-coordinate of 4 instructs us to move up 4 units from the origin, and a -4 would mean moving down.

By navigating these axes, we can locate any point and anchor geometric shapes, like the line in our equation.

Vertical Lines

Let's dive into the concept of vertical lines in a coordinate system. A vertical line runs straight up and down, which in technical terms means it is parallel to the y-axis. One critical characteristic of vertical lines is that their x-coordinates always stay constant.

If you come across an equation like our exercise where it says, for instance, \( x=4 \), what you're seeing is the blueprint for a vertical line that will slice through the x-axis precisely at the point where x is 4. It's here where the line takes a stand, defying gravity by saying, 'I shall not move left or right; upward and downward is the way I go!'

• All points on this line will share the same battle cry: 'Our x is 4!'
• No matter how high or low you go on this line, the x-coordinate is a resolute 4, while the y-coordinate can be any number — 2, -7, 15.43, doesn't matter, it's welcome on this vertical party.

Remember, vertical lines are the only types where the slope, a measure of how steep a line is, is considered undefined because you can't divide by zero.

Plotting Points

Plotting points is like a treasure hunt on the coordinate system; you're given specific coordinates, and you need to identify their location on the map. It's the basic action from which more complex graphical representations, such as lines and curves, are made. When we plot points, we translate numerical information into visual, enabling us to see patterns and relationships in data.

Here's how to ace plotting points:

• Start at the origin, the intersection point of the x-axis and y-axis, which is marked as (0,0).
• Read the coordinates. The first number tells you how many steps to take along the x-axis; the second number is the steps along the y-axis.
• Mark a dot where your horizontal and vertical travels meet. This spot is the physical manifestation of your point in the math universe.

Taking our exercise as an example, the coordinate (4,0) tells us to move 4 units to the right of the origin and stay put on the y-axis, since we're not going up or down. We now have our starting point to draw our vertical line.