Problem 18 - Algebra 1: Concepts and Skills

Question

Find the slope and y-intercept of the graph of the equation. $$y=3 x-7$$

Step-by-Step Solution

Answer

The slope of the line is 3 and the y-intercept is -7.

Step 1: Identification of Slope

The equation is in the form \(y = mx + b\). Here, \(m\) represents the slope of the line. From the equation \(y = 3x - 7\), it's clear that the slope \(m = 3\).

Step 2: Identification of y-intercept

In the same equation, \(b\) is the y-intercept. Hence, from the equation \(y = 3x - 7\), we have the y-intercept \(b = -7\).

Key Concepts

Slope

In the realm of linear equations, the concept of slope is crucial. The slope is represented by the letter \(m\) when a linear equation is written in the slope-intercept form: \(y = mx + b\).
It's a measure of how steep a line is on a graph. It tells us how much \(y\) changes for a change in \(x\). In essence, it is:

• **Rise over run** – the vertical change divided by the horizontal change between two points on the line.
• Positive if the line rises as it moves to the right.
• Negative if the line falls as it moves to the right.

In our given equation \(y = 3x - 7\), the slope \(m = 3\). This means, for every additional unit increase in \(x\), \(y\) increases by 3 units.
This positive slope indicates the line ascends steeply to the right.

y-intercept

The y-intercept of a line is where the graph intersects the y-axis. This point is crucial because it gives a starting position for the line before any movement along its path. In the slope-intercept equation \(y = mx + b\), \(b\) denotes the y-intercept.
To find the y-intercept:

• Set \(x = 0\), and calculate the corresponding \(y\) value; this value will be the y-intercept.”
• In an equation, identify the constant term; that is \(b\).

In the equation \(y = 3x - 7\), the y-intercept is \(-7\). Thus, the line crosses the y-axis at this point, indicating it begins 7 units below the origin.
This helps graph the line as it shows where the line will intersect the vertical axis without plotting extra points beforehand.

Linear Equations

Linear equations represent straight lines when graphed on a two-dimensional plane. They come in different forms, but the slope-intercept form, \(y = mx + b\), is among the most common, making it straightforward to identify key characteristics of the line.
Key features of linear equations include:

• Simplicity in representing relations between variables \(x\) and \(y\).
• Dependability for modeling constant rates of change.

The equation \(y = 3x - 7\) describes a linear relationship with a slope of 3 and y-intercept of -7. Understanding this form allows one to quickly ascertain how steep a line is and where it intersects the y-axis.
Comprehending linear equations is foundational because it aids in predicting behavior, analyzing patterns, and solving problems in algebra and calculus.