Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 22
Graph the line passing through the given point with the given slope. $$ (1,2), m=-3 $$
Short Answer
Expert verified
The line with slope -3 through (1,2) is \( y = -3x + 5 \).
Step by step solution
01
Write the Point-Slope Formula
To graph a line with a given point and slope, we use the point-slope formula of a line: \[ y - y_1 = m(x - x_1) \]where \((x_1, y_1)\) is the given point and \(m\) is the slope.
02
Substitute the Given Values
Substitute \((x_1, y_1) = (1, 2)\) and \(m = -3\) into the point-slope formula:\[ y - 2 = -3(x - 1) \]
03
Distribute the Slope
Apply the distributive property to the equation:\[ y - 2 = -3x + 3 \]
04
Solve for y
To put the equation in slope-intercept form \(y = mx + b\), solve for \(y\):1. Add 2 to both sides: \[ y = -3x + 3 + 2 \]2. Simplify: \[ y = -3x + 5 \]
05
Identify the y-intercept and Plot the Line
The line equation \(y = -3x + 5\) shows the y-intercept is 5, and the slope is \(-3\), which corresponds to a rise of -3 and a run of 1. Start at the y-intercept (0, 5) on the graph, then move down 3 units and right 1 unit to plot the next point. Draw a line through these points.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Point-Slope Formula
The point-slope formula is a very useful tool when you need to graph linear equations using a point and a slope. This formula is written as \( y - y_1 = m(x - x_1) \).
Here, \((x_1, y_1)\) represents a specific point on the line, and \(m\) is the slope.
When you have a point and a slope, this formula allows you to quickly put the line equation together:
Here, \((x_1, y_1)\) represents a specific point on the line, and \(m\) is the slope.
When you have a point and a slope, this formula allows you to quickly put the line equation together:
- Substitute the given values of the point into \((x_1, y_1)\).
- Use the slope \(m\) in the formula.
Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as \(y = mx + b\).
This format is particularly valuable because it tells you the line's slope and y-intercept directly by just looking at the equation.
In this form:
This format is particularly valuable because it tells you the line's slope and y-intercept directly by just looking at the equation.
In this form:
- \(m\) represents the slope of the line.
- \(b\) represents the y-intercept, the point where the line crosses the y-axis.
- Distributing any constants across the parentheses if necessary.
- Isolating \(y\) to one side of the equation.
Y-Intercept
The y-intercept is a crucial element in understanding linear equations. It is the point where the line crosses the y-axis on a graph. In other words, it is the value of \(y\) when \(x = 0\).
This is why in the slope-intercept form \(y = mx + b\), the \(b\) represents the y-intercept.
To find this point in a graph:
This is why in the slope-intercept form \(y = mx + b\), the \(b\) represents the y-intercept.
To find this point in a graph:
- Look at the equation in slope-intercept form to identify \(b\).
- Plot this point on the graph, generally starting as (0, \(b\)).
Slope
Slope is a fundamental concept in understanding how a linear equation represents a line. It measures the steepness and direction of the line, showing how much \(y\) changes with a shift in \(x\). This is often referred to as the "rise over run" and is represented by \(m\) in your equations.
To determine the slope:
To determine the slope:
- Start from one point on the line, calculate how far it rises or falls (change in \(y\)), and how far it runs (change in \(x\)).
- The ratio of these changes \((\text{rise}\,\,\,\text{over}\,\,\,\text{run})\) gives you the slope \(m\).
