Problem 87 - Intermediate Algebra

Question

The graph of a line passes through \((0,-3) .\) Is this the \(x\) -intercept or the \(y\) -intercept of the line?

Step-by-Step Solution

Answer

The point (0,-3) is the y-intercept.

Step 1: Understand the Intercepts

Intercepts are points where a line crosses the axes. The point where a line crosses the y-axis is called the y-intercept. Conversely, the x-intercept is the point where the line intersects the x-axis.

Step 2: Identify the Coordinates

The point given is \((0, -3)\). In a coordinate \((x, y)\), the first number \(x\) represents the x-coordinate, and the second number \(y\) is the y-coordinate.

Step 3: Analyze the Point (0, -3)

Since the x-coordinate of the point is 0 \((0, -3)\), this indicates that the line passes through the y-axis at this point. Wherever the x-value is 0, it represents the location on the y-axis, thus this point signifies the y-intercept.

Key Concepts

Understanding the y-intercept

To understand the concept of the y-intercept, imagine a coordinate plane with the x-axis running horizontally and the y-axis running vertically. The point where a line crosses the y-axis is known as the y-intercept. This is the point where the value of x is zero.
When a line intersects the y-axis, it highlights the value of y at that particular point. In mathematical terms, for any line, the equation can be written as:
\[ y = mx + c \]where \( c \) is the y-intercept. At the y-intercept, \( x = 0 \), so the equation becomes:
\[ y = c \]This means that to find the y-intercept, you simply look for the point with the format \((0, y)\).
So, in the example point (0, -3), since the x-coordinate is 0, this means the line crosses the y-axis at this point, making it the y-intercept.

Exploring the x-intercept

Contrary to the y-intercept, the x-intercept is the point where a line crosses the x-axis. Here, the y-coordinate is zero. This means x-intercepts follow the pattern \((x, 0)\).
To determine the x-intercept from an equation, set the y-value to zero and solve for x in the equation:
\[ y = mx + c \]Thus,
\[ 0 = mx + c \]This equation can be rearranged to solve for x, yielding
\[ x = -\frac{c}{m} \]So, if you are given a line equation and need to find where it crosses the x-axis, simply follow these steps. However, in the given exercise point \((0, -3)\), since the y-coordinate isn't zero, it is not the x-intercept.

Examining coordinates in algebra

Coordinates in algebra are used to specify points on a plane. Each coordinate is written as a pair \((x, y)\), where:

• \( x \) represents the horizontal position.
• \( y \) represents the vertical position.

Understanding coordinates is key in interpreting the positions of points, lines, and shapes in algebra.
When analyzing coordinates, always break them down:

• The x-coordinate shows the distance from the y-axis.
• The y-coordinate indicates the distance from the x-axis.
• If \( x = 0 \), then the point lies on the y-axis.
• If \( y = 0 \), then it lies on the x-axis.

In the exercise's point \((0, -3)\), the x-coordinate being 0 confirms its position on the y-axis, helping us determine it as the y-intercept.