Problem 19 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$x=5$$

Step-by-Step Solution

Answer

x-intercept is (5, 0). There is no y-intercept.

Step 1: Understanding the Equation

Given the equation is a vertical line, it represents that all points on this line have the same x-coordinate, which is 5.

Step 2: Finding the x-intercept

The x-intercept is where the graph crosses the x-axis. For the x-intercept, y is always 0. Therefore, substitute y = 0 into the equation. Since the equation is x = 5, the x-intercept is at the point (5, 0).

Step 3: Finding the y-intercept

The y-intercept is where the graph crosses the y-axis. For the y-intercept, x is always 0. However, since the equation is x = 5 (a vertical line), it will never cross the y-axis. Hence, there is no y-intercept for this line.

Key Concepts

Vertical Lines

In coordinate geometry, equations can represent different types of lines. One special type of line is a vertical line. Vertical lines are unique because they stretch straight up and down parallel to the y-axis. They have the same x-coordinate for all points on the line. This means that if you know the x-coordinate of a vertical line, you know that coordinate for every point on that line.
For example, the equation given in the exercise is \(x = 5\). This tells us that every point on this line has an x-coordinate of 5. Whether the point is high up or far down, the x-coordinate remains consistent.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is where we study geometric figures through a coordinate system. It involves using algebraic equations to represent geometric shapes and their properties. The most common coordinate system is the Cartesian coordinate system, which is made up of two perpendicular axes:

• The x-axis (horizontal)
• The y-axis (vertical)

In this system, every point can be described using an ordered pair \((x, y)\). The x-value tells us how far left or right the point is from the y-axis, and the y-value tells us how far up or down the point is from the x-axis.
Using equations, we can precisely describe various geometric entities:

• Lines
• Circles
• Parabolas
• And more

Intercepts

Intercepts are points where a line or curve crosses the coordinate axes. There are two types of intercepts:

• X-intercept: Where the line crosses the x-axis. At this point, the y-coordinate is zero. You find it by setting y to 0 in the equation and solving for x.
• Y-intercept: Where the line crosses the y-axis. At this point, the x-coordinate is zero. You find it by setting x to 0 in the equation and solving for y.

Given the example of the vertical line equation \(x = 5\):

• X-Intercept: To find where it crosses the x-axis, set y to 0. Since the equation does not depend on y, the line crosses the x-axis at the point (5, 0).
• Y-Intercept: To find where it crosses the y-axis, set x to 0. However, the line \(x = 5\) is vertical and never crosses the y-axis, so there is no y-intercept.