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Magazine Chapter 3: Problem 4
Graph each linear equation. \(3 x+y=9\)
Short Answer
Expert verified
Graph the line through points \((0, 9)\) and \((1, 6)\).
Step by step solution
01
Rearrange the Equation into Slope-Intercept Form
To graph the equation, first rearrange it into the slope-intercept form, which is \(y = mx + b\). Start with the given equation: \(3x + y = 9\). Solve for \(y\) by subtracting \(3x\) from both sides:\[ y = -3x + 9 \]Now, the equation is in the form \(y = mx + b\), where \(m = -3\) and \(b = 9\).
02
Identify the Slope and Y-Intercept
From the slope-intercept form \(y = -3x + 9\), identify the slope \(m\) and the y-intercept \(b\). The slope \(m = -3\) means that for each unit increase in \(x\), \(y\) decreases by 3. The y-intercept \(b = 9\) is the point at which the line crosses the y-axis.
03
Plot the Y-Intercept
Plot the y-intercept \((0, 9)\) on the graph. This is the point where the line will cross the y-axis.
04
Use the Slope to Plot a Second Point
Starting from the y-intercept \((0, 9)\), use the slope \(-3\) to determine another point on the line. From \((0, 9)\), move 1 unit to the right (positive x-direction) and 3 units down (negative y-direction), arriving at the point \((1, 6)\). Plot this point on the graph.
05
Draw the Line
Use a ruler to draw a straight line through the two points \((0, 9)\) and \((1, 6)\). This line represents the graph of the equation \(3x + y = 9\). Extend the line across the graph to show the entire linear relationship.
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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 useful ways to represent a line in algebra. It is written as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. This form allows us to easily see and understand the behavior of the line just by looking at the equation. Knowing the slope-intercept form makes graphing linear equations much simpler because you instantly know the starting point and how steep the line is.
- Rearranging: To convert an equation like \( 3x + y = 9 \) into slope-intercept form, solve for \( y \). Subtract \( 3x \) from both sides to isolate \( y \), resulting in \( y = -3x + 9 \).
- Identifying Components: In \( y = -3x + 9 \), the slope \( m \) is \(-3\) and the y-intercept \( b \) is \(9\).
Y-Intercept
The y-intercept of a line is a key point. It's where the line crosses the y-axis. In the equation \( y = mx + b \), the y-intercept is represented by \( b \). It's the value of \( y \) when \( x = 0 \). This makes it a vital starting point when graphing because it anchors the line on the graph. For the equation \( y = -3x + 9 \), the y-intercept is \( 9 \). Here鈥檚 how you locate it:
- Set \( x = 0 \): Plugging in \( x = 0 \) gives \( y = 9 \).
- Graph Position: Place a point on the graph at \( (0, 9) \).
Plotting Points
Plotting points is essential for visualizing a line. Once you have the y-intercept, use the slope to find another point. Each point on the graph represents a set of coordinates that satisfies the linear equation. With the slope intercept form, plotting becomes straightforward as you have clear guidelines:
- Start with the Y-Intercept: Place your first point at the y-intercept, for example \((0, 9)\).
- Slope Usage: Use the slope \( -3 \) to determine another point. From \((0, 9)\), move 1 unit right and 3 units down to \((1, 6)\), because the slope is \( \text{slope} = \text{rise/run} = -3/1 \).
- Additional Points: Continue using the slope to find more points if needed.
Slope of a Line
The slope of a line defines its steepness and its direction. In the equation \( y = mx + b \), \( m \) stands for the slope. It tells us how the line moves:
- Positive Slope: Means the line rises as it moves from left to right.
- Negative Slope: Indicates the line falls as you go from left to right. In our example \( m = -3 \), so the line descends three units for each unit it moves right.
- Zero Slope: Represents a horizontal line.
- Undefined Slope: Corresponds to a vertical line.
