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Chapter 0: Problem 23

Find the slope and \(y\) -intercept. $$ 3 x-3 y+6=0 $$

Short Answer

Expert verified
The slope is 1 and the \(y\)-intercept is 2.

Step by step solution

01

Identify the Equation Form

Given the equation is in the form \(3x - 3y + 6 = 0\). To find the slope and \(y\)-intercept, we need it in slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept.
02

Rearrange the Equation

Start by moving the \(x\)-term and constant to the other side of the equation: \(-3y = -3x - 6\).
03

Solve for \(y\)

Divide all terms by \(-3\) to isolate \(y\): \[y = x + 2\].Now, the equation is in slope-intercept form.
04

Identify the Slope and \(y\)-Intercept

In the equation \(y = x + 2\), the coefficient of \(x\) is the slope \(m\), which is 1, and the constant term is the \(y\)-intercept \(b\), which is 2.

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Key Concepts

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

Equation Rearrangement
To effectively solve equations and find characteristics such as slope and y-intercept, it often requires rearranging the equation. Consider the given equation \(3x - 3y + 6 = 0\). This is not yet in a form that allows easy identification of the slope or y-intercept, which is why we need to rearrange it into the slope-intercept form, \(y = mx + b\). This form makes it straightforward to extract necessary values.

Start by isolating the \(y\) variable on one side of the equation. Here's how it's done:

  • Subtract \(3x\) from both sides: \(-3y = -3x + 6\)
  • To solve for \(y\), divide every term by \(-3\): \(y = x + 2\)
After these steps, the equation is rearranged into \(y = x + 2\), a clean, easy-to-read format where we can easily pick out our slope and y-intercept values.
Slope Calculation
Calculating the slope of a linear equation in slope-intercept form \(y = mx + b\) is quite straightforward. The slope is denoted by \(m\), which represents the ratio of the change in \(y\) over the change in \(x\). In the rearranged form of the original equation, \(y = x + 2\), the coefficient of \(x\) is 1.

What does this mean for the line? Simply put:
  • For every one unit increase in \(x\), \(y\) increases by 1 unit as well.
  • This indicates a linear relationship where the line rises consistently and without turning.
Understanding the slope as a ratio helps us visualize its impact on the line's trend. The slope is a measure of the line's steepness and direction.
Y-Intercept Identification
Identifying the y-intercept in the slope-intercept form of an equation, \(y = mx + b\), is like finding a starting point for a line on a graph at the point where it crosses the \(y\)-axis. Within the equation, the y-intercept is represented by \(b\).

Considering our reorganized equation \(y = x + 2\), notice the constant term, 2.

  • This value, 2, indicates where the line crosses the \(y\)-axis.
  • On a graph, this happens when \(x = 0\).
By identifying \(b = 2\), it informs us not only about where the graph starts vertically but also gives a reference point when sketching or analyzing the behavior of the line on a coordinate plane.

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