Problem 20 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$y=4$$

Step-by-Step Solution

Answer

The y-intercept is (0, 4). There is no x-intercept.

Step 1: Identify the Equation

The given equation is a horizontal line: y = 4.

Step 2: Find the y-intercept

To find the y-intercept, set x = 0 in the equation. Since the equation doesn't depend on x, the y-intercept is simply y = 4. So, the y-intercept is (0, 4).

Step 3: Find the x-intercept

To find the x-intercept, set y = 0 in the equation. In this case, the equation y = 4 means the line never crosses the x-axis, as y is always 4 regardless of x. So, there is no x-intercept.

Key Concepts

x-intercept

To find the x-intercept of an equation, you set the y-value to zero and solve for x. The x-intercept represents the point where a graph crosses the x-axis. Think of it as the 'zero height' point of the graph.
If you're given an equation like y = mx + b, you'd set y to 0 and solve for x:
\(0 = mx + b \rightarrow x = -\frac{b}{m}\)
However, in our specific example, the equation is y = 4. Here, no matter what value of x you choose, y is always 4. So if you set y to 0, you get 0 = 4, which is not possible. That means the line never touches the x-axis, and hence, there is no x-intercept.

y-intercept

The y-intercept is the point where the graph crosses the y-axis. To find it, you set x to 0 and solve for y.
In the equation y = mx + b, you set x to 0:
\(y = m(0) + b \rightarrow y = b\)
In our case, the equation is y = 4. Here, since the equation doesn't contain any x term, it means that y is always 4 no matter what x is. So when x is 0, y is simply 4. Hence, the y-intercept is (0, 4).
In summary, the y-intercept is found directly from the equation by setting x to 0 and seeing where the line hits the y-axis.

horizontal line

A horizontal line is a special type of line where all points have the same y-coordinate. The general equation for a horizontal line is y = k, where k is a constant.
In this case, our equation is y = 4, meaning every point on the line has a y-value of 4. This results in a flat, horizontal line crossing the y-axis at (0, 4).
Horizontal lines have interesting properties:

• They have a slope of 0 because there is no vertical change as you move along the line.
• They never cross the x-axis (except when y = 0), hence no x-intercepts (in our case).

Understanding these properties helps simplify the process of finding intercepts for these lines.