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Chapter 1: Problem 8

Graph. Find the slope and the \(y\) -intercept. $$ y=3 x $$

Short Answer

Expert verified
The slope is 3, and the y-intercept is 0.

Step by step solution

01

Identify the Equation Format

The equation given is in the slope-intercept form, which is written as \[ y = mx + b \] where \(m\) represents the slope and \(b\) represents the \(y\)-intercept.
02

Determine the Slope

In the equation \( y = 3x \), compare it to the slope-intercept form. The coefficient of \(x\) is 3. Therefore, the slope \(m\) is 3.
03

Determine the y-intercept

In the equation \( y = 3x \), there is no constant term added to 3x, implying the \(y\)-intercept \(b\) is 0.
04

Summarize the Findings

The slope of the equation \( y = 3x \) is 3, and the \(y\)-intercept is 0.

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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 is a common way to express a linear equation. It is given by the formula:
$$ y = mx + b $$
Here, m stands for the slope, which tells us how steep the line is. The b represents the y-intercept, the point where the line crosses the y-axis.

For example, let's consider this equation:
$$ y = 3x $$
Notice that it's already in the slope-intercept form. By comparing it with $$ y = mx + b $$, we can easily identify the slope (m) and the y-intercept (b). Understanding this form helps to quickly graph any linear equation.
slope
The slope of a line is a measure of its steepness. It is calculated as the ratio of the vertical change to the horizontal change between two points on the line.

Mathematically, it is represented by:
$$ m = \frac{\text{rise}}{\text{run}} $$
In our example equation $$ y = 3x $$, the slope (m) is 3. This means for every 1 unit that we move to the right (run), the value of y increases by 3 units (rise). The slope tells us the direction of the line as well. A positive slope means the line ascends from left to right, while a negative slope descends.
y-intercept
The y-intercept is where the line crosses the y-axis. This value is represented by b in the slope-intercept form of the equation:

$$ y = mx + b $$
For the equation $$ y = 3x $$, we can see that there is no constant term added. Hence, the y-intercept (b) is 0.
This tells us that when x equals zero, y also equals zero. The point (0,0) is where the line intersects the y-axis. Knowing the y-intercept helps to anchor our graph and makes it easier to plot the line accurately.
graphing
Graphing a linear equation involves plotting points on the coordinate plane and drawing a line through those points. Here's a simple guide to graphing our example equation:

    \t
  • Start with the y-intercept. For $$ y = 3x $$, the y-intercept is 0. Plot this point where the line crosses the y-axis, which is at (0,0).

    • \t
  • Use the slope to find another point on the line. Since the slope (m) is 3, it means we rise 3 units up for every 1 unit we run to the right. From (0,0), move right 1 unit to (1,0) and then up 3 units to (1,3). Plot this second point.

    • \t
  • Draw a straight line through these two points, extending it in both directions. This line represents the equation $$ y = 3x $$.

Graphing lines helps to visually understand the relationship between variables in a linear equation.

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