Problem 55 - Algebra 1

Question

Write an equation in standard form of the horizontal line and the vertical line that pass through the point. $$(-4,4)$$

Step-by-Step Solution

Answer

The equation of the horizontal line is \(y = 4\) and the equation of the vertical line is \(x = -4\).

Step 1: Formulate the Equation for the Horizontal Line

For the horizontal line passing through the point (-4,4), the y-coordinate is constant. Therefore, the equation of the line is \(y = b\), where \(b\) is the y-coordinate of the given point. Here, \(b = 4\), so the horizontal line's equation that passes through (-4,4) is \(y = 4\).

Step 2: Formulate the Equation for the Vertical Line

For the vertical line passing through the point (-4,4), the x-coordinate is constant. Therefore, the equation of the line is \(x = a\), where \(a\) is the x-coordinate of the given point. In this case, \(a = -4\), so the vertical line's equation that passes through (-4,4) is \(x = -4\).

Key Concepts

Horizontal Line Equation

When you think about horizontal lines, picture a ruler laying flat on a table. These types of lines run from left to right and never change their vertical position. Hence, their slope is always zero. The key feature of horizontal lines is that they maintain a constant y-coordinate, no matter the value of x.

To write the equation for a horizontal line, you use the form:

• \(y = \text{(constant)}\)

This means wherever you are on the line, y remains unchanged. So, if a horizontal line passes through a point like \((-4, 4)\), the equation is simply \(y = 4\) because the vertical position is always 4. This doesn't change even if x moves left or right on the number line.

Vertical Line Equation

Vertical lines are like tall trees standing straight up. These lines run up and down, without any horizontal shift. Unlike horizontal lines, vertical lines have an undefined slope because their x-coordinate remains constant.

The equation of a vertical line is given by:

• \(x = \text{(constant)}\)

This expression suggests that no matter how high or low you are on the line, x stays the same. For example, if a vertical line passes through the point \((-4, 4)\), the line's equation is \(x = -4\). Here, the x-value is consistently -4, even if y varies up and down the line.

Standard Form Linear Equations

Standard form is a way of writing linear equations neatly and uniformly. The standard form of a line is written as:

\[Ax + By = C\]
where:

• A, B, and C are integers
• A is typically positive
• A and B are not both zero

When dealing with horizontal and vertical lines, we adapt this format appropriately. For a horizontal line like \(y = 4\), you can rearrange it to fit the standard form as \(0 \cdot x + 1 \cdot y = 4\), simplifying just to \(y = 4\).

Similarly, a vertical line such as \(x = -4\) can be expressed as \(1 \cdot x + 0 \cdot y = -4\), simplifying to \(x = -4\). These arrangements align with the characteristics of the standard form, even if they initially seem out of the ordinary compared to sloped lines.