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Chapter 5: Problem 60

Review graphing linear equations (Chapter 3). Graph. $$8 x=4 y[3.2]$$

Short Answer

Expert verified
Rewrite as \(y = 2x[3.2]\), plot y-intercept \([3.2]\), use slope to find another point, draw the line.

Step by step solution

01

Rewrite the Equation

Rewrite the given equation in slope-intercept form, which is \(y = mx + b\). Start by isolating \(y\). The given equation is \(8x = 4y[3.2]\). Divide both sides by 4 to simplify: \(2x = y[3.2]\). Rearrange it as \(y = 2x[3.2]\).
02

Interpret the Equation

The slope-intercept form equation is \(y = 2x[3.2]\), where the slope \(m\) is 2, and the y-intercept \(b\) is \([3.2]\). This means the line crosses the y-axis at \([3.2]\) and has a slope of 2.
03

Plot the Y-Intercept

On a graph, plot the y-intercept \([3.2]\). This is the point where the line crosses the y-axis.
04

Use the Slope to Find Another Point

The slope is 2, which means rise over run is \(2/1\). From the y-intercept \([3.2]\), go up 2 units and to the right 1 unit to find another point on the line. Plot this point.
05

Draw the Line

Draw a straight line through the points plotted. This is the graph of the equation \(8x = 4y[3.2]\).

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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 of a linear equation is one of the most common ways to express a linear equation. It is given as: \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept. This form is very useful because it immediately tells us two important pieces of information about the line: how steep it is (the slope) and where it crosses the y-axis (the y-intercept). By using the slope-intercept form, we can easily graph any linear equation with just a couple of steps.
slope
The slope of a line describes how steep the line is. It is calculated as the change in the y-values divided by the change in the x-values (often referred to as 'rise over run'). In the equation \(y = mx + b\), the slope is represented by \(m\). For example, in the equation \(y = 2x + 3.2\), the slope \(m\) is 2. This means that for every 1 unit you move to the right along the x-axis, the y-value increases by 2 units. Knowing the slope helps us understand the direction and steepness of the line.
y-intercept
The y-intercept is the point where the line crosses the y-axis. In the equation \(y = mx + b\), the y-intercept is represented by \(b\). This is the value of \(y\) when \(x\) is 0. For example, in the equation \(y = 2x + 3.2\), the y-intercept \(b\) is 3.2. To graph the y-intercept, simply find the point on the graph where \(y\) equals 3.2 and mark it. This helps us establish a starting point from which we can use the slope to find other points on the line.
plotting points
Plotting points is a fundamental idea in graphing linear equations. After finding the y-intercept, use the slope to find additional points. For instance, with the equation \(y = 2x + 3.2\), start at the y-intercept (0, 3.2). Then, use the slope of 2 (rise/run = 2/1) to plot the next point. From (0, 3.2), move up 2 units and 1 unit to the right to get to (1, 5.2). Mark this point on the graph. Repeat this to find more points if necessary. Finally, draw a straight line through all these points to represent the equation. This process makes it easy to visualize and understand the linear relationship.

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