Problem 50 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through \((1,-5),\) parallel to the \(y\) -axis

Step-by-Step Solution

Answer

The equation is x = 1.

Step 1: Understand the Problem

We're asked to find the equation of a line that goes through a specific point \(1, -5\) and is parallel to the y-axis. A line parallel to the y-axis is a vertical line.

Step 2: Identify the Characteristics of the Line

A vertical line has the equation form \(x = a\), where \(a\) is the x-coordinate of any point on the line. Vertical lines do not fit the slope-intercept form \(y = mx + b\), because their slope is undefined.

Step 3: Determine the Equation

Since the line is vertical and passes through the point \(1, -5\), it will intersect the x-axis at \(x = 1\). Thus, the equation of the line is simply \(x = 1\).

Key Concepts

Vertical Line

A vertical line is quite unique in the world of mathematics. Unlike most lines that might tilt left or right, a vertical line runs straight up and down the graph. The defining characteristic of a vertical line is that all points on the line have the same x-coordinate. For example, if a vertical line passes through the point (1, -5), it means every point along this line has an x-coordinate of 1. Here's a simple rundown of vertical lines:

• The equation is in the form \(x = a\), where \(a\) is the x-coordinate.
• Vertical lines have an undefined slope because you cannot divide by zero. If you try to calculate the slope between any two points on a vertical line, you'll end up dividing by zero, which is mathematically impossible.
• Vertical lines never cross the y-axis, so they do not have a y-intercept.

Understanding these basics will help you recognize and work with vertical lines in different math problems.

Slope-Intercept Form

The slope-intercept form is one of the most commonly used ways of writing the equation of a straight line. This form helps you easily identify key features of a line, namely its slope and y-intercept. The standard form for this equation is: \[y = mx + b\] In this form:

• \(m\) represents the slope of the line. The slope indicates how steep the line is and tells you how much \(y\) changes for a unit increase in \(x\).
• \(b\) is the y-intercept. This is where the line crosses the y-axis, meaning it's the value of \(y\) when \(x = 0\).

The slope-intercept form is perfect for lines that aren't vertical. However, it doesn't work for vertical lines because these have an undefined slope. That's why when dealing with vertical lines, we switch from slope-intercept form \(y = mx + b\) to simply \(x = a\). Knowing how to switch forms helps when graphing lines or solving problems that incorporate different styles of linear equations.

Equation of a Line

Finding the equation of a line is all about determining a formula that represents all the points on the line. The process varies slightly depending on whether the line is vertical, horizontal, or neither. Let's break it down a bit:

• Vertical Lines: If a line is vertical, its equation is \(x = a\), where \(a\) is the constant x-coordinate for the line. Since vertical lines don't "run" from left to right, they don't have a slope you can calculate.
• Horizontal Lines: These slip easily into the slope-intercept form as \(y = b\), where \(b\) is the constant y-coordinate. The slope of a horizontal line is zero because the line is perfectly flat.
• Other Lines: Besides vertical and horizontal lines, we use the slope-intercept form \(y = mx + b\) for most lines. This helps you quickly identify the slope and y-intercept for easy graphing.

Finding the exact equation depends on knowing whether the line is vertical, horizontal, or slanted. Each type uses its characteristics to arrive at the most accurate equation. Being able to write equations of all types of lines ensures you can tackle a wide variety of math problems with ease.