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Magazine Chapter 0: Problem 9
Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") $$8 y-2 x=4$$
Short Answer
Expert verified
Rewrite the equation as \(y = \frac{1}{4}x + \frac{1}{2}\), plot points (0, \(\frac{1}{2}\)) and (4, \(\frac{3}{2}\)), and draw the line.
Step by step solution
01
Rewrite the Equation in Slope-Intercept Form
The given equation is \(8y - 2x = 4\). Start by isolating y. First, add 2x to both sides: \[8y = 2x + 4\]. Next, divide everything by 8: \[y = \frac{1}{4}x + \frac{1}{2}\]. Now, the equation is in the slope-intercept form \(y = mx + b\), where the slope \(m = \frac{1}{4}\) and the y-intercept \(b = \frac{1}{2}\).
02
Identify the Slope and Y-Intercept
From the equation \(y = \frac{1}{4}x + \frac{1}{2}\), the slope \(m\) is \(\frac{1}{4}\) and the y-intercept \(b\) is \(\frac{1}{2}\). This means the line crosses the y-axis at the point (0, \(\frac{1}{2}\)).
03
Plot the Y-Intercept
On a graph, locate the point (0, \(\frac{1}{2}\)). This is the point where the line will cross the y-axis.
04
Use the Slope to Find Another Point
The slope \(\frac{1}{4}\) means that for every 4 units you move to the right on the x-axis, you move 1 unit up on the y-axis. From the y-intercept (0, \(\frac{1}{2}\)), move 4 units to the right to (4, \(\frac{1}{2}\)), then 1 unit up to (4, \(\frac{1}{2} + 1\)), which is (4, \(\frac{3}{2}\)). Plot this point on the graph.
05
Draw the Line
Using a ruler, draw a straight line passing through the points (0, \(\frac{1}{2}\)) and (4, \(\frac{3}{2}\)). Extend the line across the graph.
06
Label the Graph
Label the line with its equation \(y = \frac{1}{4}x + \frac{1}{2}\) to complete the graph.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope-intercept form
To make graphing easier, we first put the equation in slope-intercept form. The slope-intercept form of a linear equation is given by: \( y = mx + b \). Here, \( m \) stands for the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis. For instance, consider you have the equation \(8y - 2x = 4\). To use it for graphing, we isolate \( y \). This way:
- Add \(2x\) to both sides to get: \(8y = 2x + 4\).
- Next, divide everything by 8: \(y = \frac{1}{4}x + \frac{1}{2}\).
- Now the equation fits the form \( y = mx + b \), with \(m = \frac{1}{4} \) and \(b = \frac{1}{2} \). See how convenient and clear this form is!
slope
The slope \( m \) of a line indicates its steepness and direction. In our example, the slope is \( \frac{1}{4} \). Slope tells us how much \( y \) changes when we change \( x \) by a certain amount. In simple terms, if the slope is \( \frac{1}{4} \), then for every 4 units you move to the right on the x-axis, the line goes up by 1 unit on the y-axis.
- A positive slope means the line ascends as you move to the right.
- A negative slope means the line descends as you move to the right.
y-intercept
The y-intercept \( b \) of a line is the point where it crosses the y-axis. In our equation, it is \( \frac{1}{2} \). This means that when \( x = 0 \), \( y = \frac{1}{2} \). We can quickly locate this point on the graph.
- From the origin (0, 0), move up by half a unit.
- This location is (0, \( \frac{1}{2} \)).
plotting points
Now that we know the y-intercept and the slope, we can plot points to graph the line. Let's start with the y-intercept:
- Locate (0, \( \frac{1}{2} \) on the graph.
- Starting from (0, \( \frac{1}{2} \)), move 4 units to the right: (4, \( \frac{1}{2} \)).
- Then, move 1 unit up: (4, \( \frac{3}{2} \)).
linear equation
A linear equation represents a straight line when graphed. The general form of a linear equation is \( y = mx + b \). In our example, \( y = \frac{1}{4}x + \frac{1}{2} \), we see the characteristics of a linear equation:
- It has no exponents on y or x (both have the power of 1).
- Its graph is always a straight line.
