Problem 28 - Algebra 1: Concepts and Skills

Question

Graph the equation. $$ x=\frac{3}{4} $$

Step-by-Step Solution

Answer

The graph of \( x = \frac{3}{4} \) is a vertical line that crosses the x-axis at \( x = \frac{3}{4} \).

Step 1: Understand the Equation

The equation given is \( x = \frac{3}{4} \). This is a vertical line which crosses the x-axis at \( \frac{3}{4} \).

Step 2: Draw the Axes

Draw the x and y axes on the graph. It can be completed on any paper or digitally using a plotter. Note that the graph for \( x = a \) form does not depend on values of y-axis.

Step 3: Plot the Line

Plot a vertical line on the graph at the position \( x = \frac{3}{4} \). This line should run vertically, implying it passes through all y values at \( x = \frac{3}{4} \).

Key Concepts

Vertical Lines

When we talk about vertical lines in graphing, we're discussing lines that move up and down the coordinate plane, instead of left and right. A vertical line is represented by an equation of the form \( x = a \). This means that for every point on this line, the x-coordinate is constant.

• A vertical line runs parallel to the y-axis.
• All points on a vertical line have the same x-coordinate.
• The equation for a vertical line does not include a y-variable.

For example, with the equation \( x = \frac{3}{4} \), you're dealing with a line that crosses the x-axis exactly at \( x = \frac{3}{4} \), no matter what the y-value might be. This means that whether \( y \) is 0, 1, or -3, the x-value remains \( \frac{3}{4} \). Vertical lines are unique because they do not have a slope. You might say their slope is "undefined" because the steepness does not "move" left to right.

Coordinate Plane

The coordinate plane is a fundamental concept in graphing. It's a two-dimensional surface where you can plot points, lines, and curves to visually represent equations. It includes:

• The x-axis: a horizontal line.
• The y-axis: a vertical line.
• Four quadrants that separate the axes.

Each point on the coordinate plane is defined by a pair \((x, y)\). When graphing, the x-coordinate tells you how far you need to move horizontally from the origin (where the x-axis and y-axis cross), and the y-coordinate tells you how far to move vertically.
To graph a line like \( x = \frac{3}{4} \), you'd start at the point on the x-axis labeled \( \frac{3}{4} \) and draw a line that spans all y-values, staying parallel to the y-axis.

Equation of a Line

An equation of a line provides all the information you need to graph it on a coordinate plane. Depending on the equation's form, you can determine the key characteristics of the line, such as its slope, y-intercept, or simply the fixed x-value for vertical lines.
For most linear equations, the standard form is \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept.

However, vertical lines do not fit this form because they do not have a slope. Instead, their equation is simply \( x = a \). Here, \( a \) represents the fixed x-coordinate of the vertical line.
In the specific case of \( x = \frac{3}{4} \), it tells you that this line will not "move" horizontally. It only changes vertically, spanning all possible y-values with a constant x-value of \( \frac{3}{4} \). This kind of simplicity makes vertical lines a unique and important part of understanding line equations.