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Chapter 2: Problem 61

Find the slope of each line, and sketch its graph. \(x-3=0\)

Short Answer

Expert verified
The slope of the line is undefined. Draw a vertical line at \(x = 3\).

Step by step solution

01

Identify the Equation Form

The given equation is in the form of a vertical line: \(x - 3 = 0\)
02

Solve for 'x'

Solve for 'x' to find the value of the line: \(x = 3\)
03

Interpret the Slope

For a vertical line such as \(x = 3\), the slope is undefined because the change in 'y' can be anything while 'x' does not change.
04

Sketch the Graph

Draw a vertical line on the graph at \(x = 3\). This line extends infinitely up and down on the y-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertical line equation
A vertical line equation is one of the simplest forms of linear equations. It looks like this: \(x = a\).
In this type of equation, 'a' is a constant. This means that for every point on this line, the value of 'x' is always 'a'. The 'y' value can change and represent any number, stretching infinitely up and down.
For example, if we have \(x - 3 = 0\), we can solve it to get \(x = 3\). This means on the graph, there's a straight vertical line passing through \(x = 3\).
  • Constant 'a' fixes the x-coordinate.
  • The line is vertical, parallel to the y-axis.
Vertical lines are unique in that they exhibit special properties, especially related to their slope and graphing.
undefined slope
Understanding an undefined slope is critical when dealing with vertical lines. Slope represents how steep a line is. It's calculated by how much 'y' changes compared to 'x'. When we look at a vertical line, the value of 'x' does not change, no matter how much 'y' changes.
This means we would be dividing by zero in the slope formula, which is \(\frac{\text{change in } y}{\text{change in } x}\). Division by zero is undefined in mathematics, making the slope of any vertical line undefined.
  • Vertical lines' slopes are always undefined.
  • It occurs because of division by zero.
So, for a vertical line like \(x = 3\), there's no number to represent its slope - it's simply undefined.
graphing lines
Graphing lines, especially vertical ones, can be straightforward. Here鈥檚 how you can plot a vertical line like \(x = 3\) effectively:
1. Identify 'x' value: In our equation \(x = 3\), we know that the line passes through 3 on the x-axis.
2. Draw the Line: Since 'x' is always 3, draw a straight vertical line crossing the x-axis at 3.
  • The line will run parallel to the y-axis.
  • It will extend infinitely both upwards and downwards.
When graphing lines, always start with the intercepts. For vertical lines, knowing the 'x' value is enough since this line does not depend on 'y' values. This makes graphing vertical lines particularly simple and direct.

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