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Magazine Chapter 2: Problem 130
For the following exercises, sketch the graph of each equation. $$4 x=-9 y+36$$
Short Answer
Expert verified
The graph is a straight line with a slope of \(\frac{4}{9}\) crossing the y-axis at \(-4\).
Step by step solution
01
Rewrite the Equation in Slope-Intercept Form
To sketch the graph, we start by rewriting the equation in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Starting with the equation \(4x = -9y + 36\), we rearrange it to solve for \(y\). First, move \(9y\) to the other side by adding \(9y\) to both sides: \(4x + 9y = 36\). Now, solve for \(y\) by isolating it: \(9y = 4x - 36\). Finally, divide everything by 9: \(y = \frac{4}{9}x - 4\).
02
Identify the Slope and Y-intercept
From the equation \(y = \frac{4}{9}x - 4\), we can identify the slope \(m\) as \(\frac{4}{9}\) and the y-intercept \(b\) as \(-4\). This tells us the line rises 4 units for every 9 units it moves horizontally, and it crosses the y-axis at \(y = -4\).
03
Plot the Y-intercept
Begin sketching the graph by plotting the y-intercept on the graph. The y-intercept is the point where the graph crosses the y-axis, which is \((0,-4)\) in this case. Mark this point on the graph.
04
Use the Slope to Plot a Second Point
Using the slope \(\frac{4}{9}\), move from the y-intercept up 4 units and right 9 units to plot a second point, \((9, 0)\). This ensures that our graph will correctly reflect the slope.
05
Draw the Line
With the two points \((0, -4)\) and \((9, 0)\) plotted, draw a straight line through these points. This line represents the graph of the equation \(4x = -9y + 36\).
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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 is a common way to represent linear equations and allows for easy graphing of a line. This form is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. This form is incredibly useful because it clearly shows both the direction and steepness of the line (from the slope) and where the line crosses the y-axis (at the y-intercept).
To graph a line using this form, simply identify the slope and the y-intercept. With these two pieces of information, you can easily draw the line on a coordinate plane by starting at the y-intercept and following the direction given by the slope.
To graph a line using this form, simply identify the slope and the y-intercept. With these two pieces of information, you can easily draw the line on a coordinate plane by starting at the y-intercept and following the direction given by the slope.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This point has coordinates \((0, b)\), since it occurs at \(x = 0\). The y-intercept is a key starting point for graphing a line because it gives you one point that lies directly on the line.
- In the equation \(y = \frac{4}{9}x - 4\), the y-intercept \(b\) is \(-4\).
- To plot this, find \(-4\) on the y-axis and mark a point there.
Plotting Points
Plotting points is an essential skill for graphing lines and interpreting linear equations. Once you have the y-intercept, you can find additional points using the slope. By plotting at least two points, you can accurately draw a line.
- Start by plotting the y-intercept. This is always your first point.
- Use the slope to find a second point. The slope \(\frac{4}{9}\) means for every 9 units you move to the right, go up 4 units.
- From the point \((0, -4)\), move right 9 units and up 4 units to plot the point \((9, 0)\).
Slope of a Line
The slope of a line indicates how steep the line is, and it is usually expressed as a ratio of the vertical change to the horizontal change between any two points on the line. The slope is denoted by \(m\) in the slope-intercept form \(y = mx + b\).
The slope can be positive, negative, zero, or undefined, depending on the direction of the line:
The slope can be positive, negative, zero, or undefined, depending on the direction of the line:
- A positive slope (\(m > 0\)) means the line rises from left to right.
- A negative slope (\(m < 0\)) means the line falls from left to right.
- A zero slope (horizontal line) means there is no vertical change.
- An undefined slope (vertical line) means there is no horizontal change.
