Problem 26 - Algebra 1: Concepts and Skills

Question

Graph the equation. $$ x=4 $$

Step-by-Step Solution

Answer

The graph of the equation \(x=4\) is a vertical line passing through the x-axis at the point where \(x\) equals 4.

Step 1: Identify the Type of Line

The given equation is \(x=4\). When an equation occurs where \(x\) is equal to a constant, it means that it is a vertical line. In this case, every point on this line has an x-coordinate of 4.

Step 2: Drawing the Line

To graph this line, one needs to draw a vertical line through the point where \(x\) is 4 on the x-axis.

Step 3: Verification

Any point on the line should satisfy the equation. This means that the x-coordinate should be 4 for every point as prescribed by the equation. Check a few points on the line to make sure this is true.

Key Concepts

Vertical Line

A vertical line on the coordinate plane is quite easy to understand because it has a unique characteristic. It is a line where all the points have the same x-coordinate. This means that no matter how far up or down you go along the line, the x-value remains constant. In the case of the equation \(x = 4\), every point along the line has the first coordinate (x-coordinate) as 4. This creates a line that extends vertically in both directions through the point.

Vertical lines are special because they do not have a slope. The concept of slope involves a change in y relative to a change in x, and with a vertical line, such changes in x do not happen – it remains constant. As a result, the slope of a vertical line is undefined.

To summarize, when dealing with equations like \(x = a\) where 'a' is a constant, you are graphing a simple vertical line through the x-axis at that specific point.

Equation of a Line

Understanding the equation of a line forms the foundation for graphing and interpreting linear equations. Typically, a line's equation can be represented in the form \(y = mx + b\) where 'm' is the slope and 'b' is the y-intercept.

However, vertical lines are a unique exception to this formula. For them, the equation takes the form \(x = a\), where 'a' is a constant number. The line is defined by all points having the same x-coordinate and is entirely independent of the y-coordinate. This is why vertical lines don’t intersect the y-axis as other linear equations do.

For example, in the equation \(x = 4\), all points (4, y) will lie on the line. Here, 'y' can be any real number, but every x will always be 4. This makes the line a straight vertical through x = 4 on the graph.

Coordinate Plane

The coordinate plane is an essential tool for graphing equations such as \(x = 4\). It consists of two perpendicular lines, known as axes:

• The horizontal axis, called the x-axis
• The vertical axis, called the y-axis

These axes divide the plane into four quadrants.

Coordinates are written as pairs (x, y) and represent points on this plane. The origin, located at (0, 0), is where the x-axis and y-axis intersect. When graphing a vertical line, you start by identifying the x-coordinate for the line. In our example, each point on the line \(x = 4\) has an x-coordinate of 4.

From this, you draw a line parallel to the y-axis at the x-value of 4. This results in a vertical line which serves as a simple visual representation of all the points that fulfill the condition \(x = 4\) on the coordinate plane.