Problem 51 - Elementary Algebra

Question

For Problems \(45-60\), write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((5,6)\) and is perpendicular to the \(y\) axis

Step-by-Step Solution

Answer

The equation is \( y = 6 \) or \( 0x + 1y = 6 \).

Step 1: Identify the Line Perpendicular to the Y-Axis

A line that is perpendicular to the y-axis is a horizontal line. This means it has a constant y-value and can be expressed as \( y = c \), where \( c \) is a constant.

Step 2: Determine the Constant Using the Given Point

We are given that the line contains the point \((5,6)\). Since the line is horizontal, its y-value is always 6. Therefore, the equation of the line is \( y = 6 \).

Step 3: Convert the Equation to Standard Form

To express the equation in standard form, we want an equation of the form \( Ax + By = C \). For the horizontal line \( y = 6 \), this can be rearranged as \( 0 \cdot x + 1 \cdot y = 6 \), which simplifies to \( y = 6 \). The standard form of this equation is \( 0x + 1y = 6 \).

Key Concepts

Perpendicular to Y-Axis

When a line is perpendicular to the y-axis, it means that the line forms a 90-degree angle with the vertical direction on a graph. Such a line is always a horizontal line. To visualize this, imagine standing in front of a straight wall (the y-axis) and stretching your arms out straight to the sides. That horizontal line formed by your arms is perpendicular to the wall.

In mathematical terms, a line that is perpendicular to the y-axis does not change its y-coordinate as it moves left or right. This means it has a constant y-value. For instance, if the line passes through the point \(5, 6\), then the y-coordinate is 6 for all x-values. Consequently, the equation of this line is simply \(y = 6\).

It's important to remember:

• Horizontal lines are always perpendicular to the y-axis.
• All points on such a line have the same y-coordinate.

Horizontal Line

A horizontal line is a straight line that runs left to right, parallel to the x-axis. It has zero slope, meaning there is no vertical change as you move along the line. If you think of a calm, flat horizon, that perfectly represents a horizontal line.

Mathematically speaking, a horizontal line's equation is \(y = c\), where \(c\) is a constant. This indicates that no matter what the x-coordinate is, the y-coordinate remains unchanged.

For example, if you are told that a line is horizontal and passes through the point \(5, 6\), the y-value for every point on this line would be 6. Hence, you can confidently write the line's equation as \(y = 6\).

Key points include:

• The slope of a horizontal line is 0.
• The line is expressed in the form \(y = c\).

Equation of a Line

In mathematics, the equation of a line is a way to describe a straight line using algebraic terms. Generally, the line's equation can be found in several forms, including slope-intercept form and standard form.

For a horizontal line, the equation is exceptionally straightforward. It is given by \(y = \) some constant, reflecting that every point on the line shares the same y-coordinate. However, to express this equation in standard form, which is written as \(Ax + By = C\), adjustments might be necessary.

For instance, the line \(y = 6\) can be transformed to standard form by recognizing that there is no x-component. So it becomes \(0x + 1y = 6\). Despite the inclusion of a zero for the x-term, this still perfectly adheres to the standard form's requirements.

Remember:

• Standard form adds clarity and uniformity when comparing different linear equations.
• It makes algebraic manipulation easier in many scenarios.