Problem 56 - Algebra 1

Question

Write an equation in standard form of the horizontal line and the vertical line that pass through the point. $$(-2,-5)$$

Step-by-Step Solution

Answer

The equation for the horizontal line is \(y = -5\) and the equation for the vertical line is \(x = -2\).

Step 1: Find equation for the horizontal line

The horizontal line that passes through the point (-2,-5) has a constant \(y\) value which is -5. Therefore, the equation for this line is \(y = -5\).

Step 2: Find equation for the vertical line

The vertical line passing through the same point has a constant \(x\) value which is -2. Therefore, the equation for this line is \(x = -2\).

Key Concepts

Equations of Lines

To express a line mathematically, we need to understand its nature: a line is a straight path that extends infinitely in both directions. One common method to describe a line is through an equation. There are several forms that these equations can take, including standard, slope-intercept, and point-slope forms.

- **Standard form:** Typically written as \(Ax + By = C\), where \(A, B,\) and \(C\) are integers. The standard form has clarity advantages in many cases, especially when dealing with systems of equations.- **Slope-intercept form:** Written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. It is often used for its straightforward representation of a line's steepness and starting point.- **Point-slope form:** Useful in specific calculations, written as \(y - y_1 = m(x - x_1)\), making it ideal for lines you want to draw or calculate from a given point \((x_1, y_1)\) with slope \(m\).These variations are just different ways to express the same straight line, emphasizing different aspects such as slope or starting coordinates. It's crucial to understand which form offers the most advantage for the situation you're analyzing.

Horizontal and Vertical Lines

Horizontal and vertical lines have unique features in geometry, generally because they follow strict patterns:- **Horizontal Lines**: These lines run left to right (or right to left) and remain parallel to the x-axis. They have an equation of the form \(y = c\), where \(c\) is a constant. Notably, for horizontal lines, the slope is 0, making them flat.

- **Vertical Lines**: These lines run up and down and are parallel to the y-axis. Their equation is \(x = c\), where \(c\) is the constant x-value for all points on the line. Vertical lines have an undefined slope, which arises from attempting to divide by zero during slope calculations.

Recognizing these types of lines and their equations helps when graphing or interpreting line data in a coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we graphically represent points, lines, and curves. It consists of two axes:- **x-axis:** Runs horizontally, representing all possible x-coordinate values. Think of it like the ground level in a graph.- **y-axis:** Runs vertically, representing y-coordinate values. Playing a role similar to height in a graph.

Every point on the coordinate plane is represented by a pair \((x, y)\), where \(x\) denotes horizontal position, and \(y\) denotes vertical position. These axes intersect at a point called the origin (0, 0).

Being familiar with the coordinate plane framework is crucial for interpreting and creating graphs. It serves as the foundation for plotting lines like the ones we've discussed, including those horizontal and vertical lines that can be defined by simple coordinate equations.