Problem 60 - Algebra 1

Question

Write an equation in standard form of the horizontal line and the vertical line that pass through the point. $$(-3,7)$$

Step-by-Step Solution

Answer

The standard form of the horizontal line passing through (-3,7) is \(y=7\) and that of the vertical line is \(x=-3\).

Step 1: Find the equation of the horizontal line

In a horizontal line, the y-coordinate stays the same. Since our point is (-3,7), the y-coordinate is 7. Replace 'k' with 7. Hence the equation of the horizontal line passing through (-3,7) is \(y=7\).

Step 2: Find the equation of the vertical line

In a vertical line, the x-coordinate stays the same. Since our point is (-3,7), the x-coordinate is -3. Replace 'h' with -3. Hence, the equation of the vertical line passing through (-3,7) is \(x=-3\).

Key Concepts

Horizontal Line

A horizontal line is a straight line that extends left and right on a graph. It does not rise or fall; it remains constant in the vertical direction. The defining characteristic of a horizontal line is that the

• Y-coordinate remains constant for any point on the line.
• The slope of a horizontal line is always zero.

This means the equation of a horizontal line is written as \(y = k\), where \(k\) represents the constant y-value for every point on the line. For instance, if the line goes through the point \((-3, 7)\), the equation of the horizontal line is \(y = 7\). It means that no matter which x-value you choose, the y-value will always be 7.
Horizontal lines are one of the two simplest linear equations and are vital in understanding basic functions.

Vertical Line

Vertical lines are straight lines moving up and down, and they do not extend horizontally. Their unique feature is that:

• The X-coordinate remains constant for any point on the line, no matter the value of y.
• Vertical lines are undefined in slope because the rate of change is infinite; it never changes horizontally.

The equation of a vertical line is expressed as \(x = h\), where \(h\) is the constant x-value for the line. For example, if the line passes through \((-3, 7)\), the vertical line's equation is \(x = -3\). It denotes that regardless of the y-value, the x-value remains at -3.
Vertical lines are integral to coordinate geometry and help form a solid foundational understanding of graphing linear equations.

Standard Form

The standard form of a linear equation is an expression where all terms are expressed in a specific configuration, typically written as \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative. This form is immensely useful for algebraic calculations, offering a structured way to present lines especially when dealing with their intersections.
Although horizontal and vertical lines don't always fit the classic two-variable linear form, they can be adjusted into the form \(Ax + By = C\):

• Horizontal lines can be represented as \(0x + By = C\), which simplifies to \(y = C/B\).
• Vertical lines can be illustrated as \(Ax + 0y = C\), simplifying to \(x = C/A\).

When learning about different types of lines, understanding how they connect to the concept of standard form can provide a strong mathematical foundation. It allows for clearer communication and easier problem-solving.