Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 35
Graph each linear equation. See Examples 1 through 7. $$ x+8 y=8 $$
Short Answer
Expert verified
Rewritten as \(y = -\frac{1}{8}x + 1\), it graphs with a y-intercept at (0, 1) and slope -1/8.
Step by step solution
01
Rewrite in Slope-Intercept Form
The first step is to rewrite the equation into the slope-intercept form, which is given by \(y = mx + b\). Start with the original equation: \(x + 8y = 8\). To isolate \(y\), subtract \(x\) from both sides: \[8y = -x + 8\]Then, divide each term by 8 to solve for \(y\): \[y = -\frac{1}{8}x + 1\].This is our slope-intercept form, where the slope \(m = -\frac{1}{8}\) and the y-intercept \(b = 1\).
02
Identify the Slope and Y-Intercept
From the equation \(y = -\frac{1}{8}x + 1\), we identify two key values:- The slope \(m = -\frac{1}{8}\), which tells us the rise over run.- The y-intercept \(b = 1\), which is the point \((0, 1)\) on the graph.
03
Plot the Y-Intercept
Start graphing by plotting the y-intercept on the graph. This is where the line crosses the y-axis. The point is at \((0, 1)\). Mark this point on the graph.
04
Use the Slope to Find Another Point
The slope of the line is \(-\frac{1}{8}\), meaning the line falls 1 unit vertically for every 8 units it moves horizontally to the right. From the y-intercept \((0, 1)\), move down 1 unit to \((0, 0)\) (since it鈥檚 a drop) along the y-axis and then right 8 units to \((8, 0)\). Plot this second point \((8, 0)\) on the graph.
05
Draw the Line
Now that you have two points \((0, 1)\) and \((8, 0)\), use a ruler to connect them in a straight line. Extend the line across the graph, ensuring it crosses the y-axis at \((0, 1)\) and through the second point \((8, 0)\). This is the graph of the equation \(x + 8y = 8\).
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.
Slope-Intercept Form
The slope-intercept form is a method for expressing linear equations. This form is especially helpful for identifying important parts of a line, like its slope and y-intercept. The formula for the slope-intercept form is written as: \[y = mx + b\]Here, the letter \(m\) represents the slope of the line. The slope indicates how steep the line is, showing the rise over run.
\(b\), on the other hand, stands for the y-intercept, which is where the line crosses the y-axis.
\(b\), on the other hand, stands for the y-intercept, which is where the line crosses the y-axis.
- Knowing the slope-intercept form makes it very easy to graph a line, as it provides immediate insight into the direction and starting point of the graph.
- Once an equation is rearranged into this form, plotting it becomes much more straightforward.
Graphing Linear Equations
Graphing linear equations involves representing equations as lines on a graph. To do this correctly, you mainly need the slope and y-intercept. Once you have these, it becomes quite intuitive.
First, identify the y-intercept, or where the line crosses the y-axis. This is the point that tells you your first mark on the graph.
First, identify the y-intercept, or where the line crosses the y-axis. This is the point that tells you your first mark on the graph.
- Then, using the slope, you determine the direction and steepness of the line.
- Your task is to find another point using the slope, which tells you how far to move up or down and right from your starting point.
- With two points plotted, you just draw a straight line through them. This represents the equation graphically.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis of a graph. It's a crucial part of understanding a linear equation and is denoted by \(b\) in the slope-intercept formula.
For the equation \[y = -\frac{1}{8}x + 1\], the y-intercept is \(1\). This means that at \(x = 0\), the value of \(y\) is \(1\). You plot this point as your starting point on the graph.
For the equation \[y = -\frac{1}{8}x + 1\], the y-intercept is \(1\). This means that at \(x = 0\), the value of \(y\) is \(1\). You plot this point as your starting point on the graph.
- The y-intercept simplifies the process of plotting because it gives a clear initial position for the line on the graph without needing to perform additional calculations.
- It also provides a point of reference to describe how other solutions given by the equation relate to the graph itself.
Slope
The slope of a line gives information about its steepness and direction. It shows how much \(y\) changes for a change in \(x\). This is termed as "rise over run."
For example, in the equation \[y = -\frac{1}{8}x + 1\], the slope is \(-\frac{1}{8}\). This means for every 8 units you move horizontally to the right, the line falls by 1 unit vertically.
For example, in the equation \[y = -\frac{1}{8}x + 1\], the slope is \(-\frac{1}{8}\). This means for every 8 units you move horizontally to the right, the line falls by 1 unit vertically.
- A positive slope means the line rises as it moves from left to right, and a negative slope means it falls.
- If the slope equals zero, the line is horizontal, indicating no change in y as x changes.
- The larger the absolute value of the slope, the steeper the line.
