Problem 61 - Algebra 1

Question

Write an equation in standard form of the horizontal line and the vertical line that pass through the point. $$(10,-3)$$

Step-by-Step Solution

Answer

The standard form for the horizontal line passing through the point (10,-3) is \(y=-3\) and for the vertical line it's \(x=10\).

Step 1: Equation for Horizontal Line

Start with the general form of a horizontal line, which is \(y=a\). Replace the \(a\) with the y-value of the given point. For the provided point (10, -3), the y-value is -3. Therefore, the horizontal line that passes through this point will have the equation \(y=-3\).

Step 2: Equation for Vertical Line

Now, let's ascertain the equation of the vertical line that passes through the provided point. A vertical line follows the equation \(x=b\). Here, \(b\) should be replaced by the x-value of the point, which in this case is 10. Consequently, the vertical line passing through the point has the equation \(x=10\).

Key Concepts

Horizontal Line

When we talk about a horizontal line in mathematics, we are referring to a straight line that runs left and right across the coordinate plane. It does not rise or fall, hence it maintains a constant y-value. The equation of a horizontal line can be written simply as \(y = c\), where \(c\) is a constant.

• For example, for a line to pass through the point \((10, -3)\), it should have a consistent y-value of \(-3\).
• Therefore, the equation becomes \(y = -3\).

Horizontal lines have an essential characteristic that they are parallel to the x-axis of the coordinate plane. This is crucial for students to understand as it highlights that the y-coordinate remains unchanged along the length of the line. So, regardless of which x-coordinate you select on the line, the y-coordinate will always be \(-3\).

Vertical Line

Vertical lines stand in contrast to horizontal lines. They run straight up and down the coordinate plane and have a constant x-value. The equation representing a vertical line is expressed as \(x = c\), where \(c\) is the constant value for the x-coordinate.

• In our specific example, the point \((10, -3)\) has an x-value of \(10\).
• Thus, the vertical line through this point would have the equation \(x = 10\).

An attribute of vertical lines is that they are parallel to the y-axis. This means that all the points on a vertical line share the same x-value, while the y-value can differ. Unlike horizontal lines, vertical lines do not have a slope that can be calculated because their slope would be undefined due to a division by zero in the slope formula, which is not permissible.

Standard Form

The standard form of a linear equation is a way of representing lines in mathematical equations. It is often displayed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative.

• In this format, horizontal lines can be adapted as \(By = C\), since there is no x-component. For our horizontal line \(y = -3\), an equivalent, though not typical, standard form is \(0 \cdot x + 1y = -3\).
• Similarly, vertical lines can be adapted as \(Ax = C\), because there's no y-component. For our vertical line \(x = 10\), a standard form representation is \(1x + 0 \cdot y = 10\).

When equations are written in standard form, it facilitates some mathematical processes, like finding intercepts efficiently. It also gives equations a uniform look, which can be advantageous when analyzing or comparing multiple linear equations.