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Chapter 1: Problem 24

Write an equation for the line with Slope -5 and \(y\) -intercept -2

Short Answer

Expert verified
The equation of the line with slope -5 and \(y\)-intercept -2 is \(y = -5x -2\)

Step by step solution

01

Identify the given slope and \(y\)-intercept

We are given that the slope (\(m))\) = -5 and the \(y\)-intercept (\(b\)) = -2.
02

Substitute the slope and \(y\)-intercept into the slope-intercept equation

Substitute -5 for \(m\) and -2 for \(b\) in the equation \(y = mx + b\). This gives us \(y = -5x -2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the easiest and most widely used ways to express a line equation. It allows you to quickly identify the slope and the y-intercept, which play crucial roles in defining the line's direction and position. The standard format for this form is:
  • \(y = mx + b\)
Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. In practical terms, the slope-intercept form allows us to see how steep a line is and where it crosses the y-axis. Understanding the slope-intercept form is essential for graphing a line quickly, making it an indispensable tool in algebra.
Slope
The slope is a fundamental concept in understanding the behavior of a line. It tells us how steep the line is and whether it rises or falls as it moves from left to right. Mathematically, the slope is represented by \(m\) in the slope-intercept form and can be calculated using two points on a line:
  • \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)
A positive slope indicates the line rises as it moves to the right, whereas a negative slope means it falls. A steeper line has a higher absolute value of the slope. In our equation, the slope \(-5\) tells us that for every step to the right, the line drops down 5 units, showing a steep downward direction.
Y-intercept
The y-intercept is the point where a line crosses the y-axis of a graph. This is when the value of \(x\) is zero. In the slope-intercept form \(y = mx + b\), the y-intercept is expressed by \(b\). It provides essential information about where the line starts on the y-axis. Understanding the y-intercept is particularly useful when you are setting up or verifying a line graph, as it instantly tells you the line's starting point on the vertical axis. For the equation \(y = -5x - 2\), the y-intercept is \(-2\), meaning the line crosses the y-axis at -2. This gives a clear picture of the line's vertical starting point and helps in sketching the graph.

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