Problem 22 - Algebra A Combined Function

Question

Write an equation of each line. See Examples 3 and \(4 .\) Vertical; through (4,7)

Step-by-Step Solution

Answer

The equation of the line is \(x = 4\).

Step 1: Understand Vertical Lines

Vertical lines have the same x-coordinate for all points on the line. This means that the x-coordinate is constant, whereas the y-coordinate can vary.

Step 2: Identify the Given Point

We are given the point (4,7). For a vertical line, the x-coordinate of any point on the line remains constant. Therefore, from the point (4,7), the x-coordinate 4 will be used for the equation.

Step 3: Write the Equation of the Vertical Line

Since the x-coordinates are constant for a vertical line through the given point (4,7), the equation of the vertical line is simply written as \(x = 4\).

Key Concepts

Vertical Lines in Coordinate Geometry

Vertical lines are a fundamental concept in coordinate geometry. They run straight up and down on a graph, parallel to the y-axis. One unique property of vertical lines is that they do not slope horizontally.
This means that all points lying on a vertical line have the same x-coordinate.

• If you imagine a graph, a vertical line will look like a column that crosses the x-axis.
• No matter how high or low you go along the line, the x-coordinate remains unchanged.

Understanding this property helps in realizing why vertical lines are characterized primarily by their x-coordinates.

Equation of a Line

When we talk about the equation of a line in general terms, it typically refers to a mathematical expression that describes all the points on the line.
For most lines, this equation is written in the form of slope-intercept: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

However, for vertical lines, this form doesn't apply because vertical lines do not have a slope in the traditional sense (their slope is undefined). Instead, they are best described by their x-coordinate because all points on the line share this same value.
Thus, the equation of a vertical line is written as \(x = a\), where \(a\) is the constant x-coordinate.
This notation tells us everything we need to know about a vertical line's position on a graph.

X-Coordinate in Vertical Lines

The x-coordinate in a vertical line is key because it is the sole identifier of the line's equation. Unlike other lines, the y-coordinate can vary, while the x-coordinate stays constant.

• For instance, with a point (4,7), all that matters for the vertical line is the first number: 4.
• This means that no matter what the second number is, whether it's 7 or any other number, the x-coordinate remains a fixed value.

That's why the equation of a vertical line passing through the point (4,7) is simply \(x = 4\).
This simplicity makes it easy to identify and write equations for vertical lines whenever you know an x-coordinate.

Writing Equations in Algebra

Writing equations in algebra involves translating a geometrical concept into a mathematical language. Using the principles of algebra, we express the relationships between variables.

In the case of vertical lines, writing the equation is straightforward.

• We look at the given point, such as (4,7), identify the x-coordinate, and use it in the equation \(x = 4\).
• There is no need to include a y-component in the equation since the y-value can vary without affecting the line's vertical nature.

Such exercises help learners understand how simple it can be to convert geometric understanding into algebraic expressions.
This is key for mastering algebra, where translating between visual graphs and equations is often required.