Problem 35
Question
Determine the slope and y-intercept (if possible) of the linear equation. Then describe its graph. $$2 x-3 y=9$$
Step-by-Step Solution
Verified Answer
The slope of the equation \(2x - 3y = 9\) is \(\frac{2}{3}\), and the y-intercept is \(-3\). The graph of this equation is a straight line that slopes upwards and crosses the y-axis at \(y = -3\).
1Step 1: Rearrange the equation
The equation needs to be rearranged to the form \(y = mx + c\). Starting from \(2x - 3y = 9\), add \(3y\) to both sides to get \(3y = 2x - 9\). Then divide every term by 3 to isolate \(y\), yielding the equation \(y = \frac{2}{3}x - 3\).
2Step 2: Identify the slope and y-intercept
From the rearranged equation \(y = \frac{2}{3}x - 3\), it's clear that the coefficient of \(x\) is the slope \(m\), and the constant term is the y-intercept \(c\). Therefore, the slope \(m\) is \(\frac{2}{3}\) and the y-intercept \(c\) is \(-3\).
3Step 3: Describe the graph
The graph of the equation \(y = \frac{2}{3}x - 3\) is a straight line (due to it being a linear equation), which slopes upwards because the slope, \(m = \frac{2}{3}\), is positive. Its y-intercept is at \(y = -3\) on the vertical axis, which means that the line crosses the y-axis at this point.
Key Concepts
Slope of a LineY-interceptGraphing Linear Functions
Slope of a Line
In linear equations like the one considered here, the slope of a line is a crucial concept. The slope, denoted by \(m\), represents the steepness and direction of the line.
It is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate (rise over run) between any two points on the line.
The formula often used is:
This positive value indicates the line rises as it moves from left to right. A higher absolute slope value would mean a steeper line.
Conversely, a negative slope would indicate a line descends from left to right.
It is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate (rise over run) between any two points on the line.
The formula often used is:
- \( m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1} \)
This positive value indicates the line rises as it moves from left to right. A higher absolute slope value would mean a steeper line.
Conversely, a negative slope would indicate a line descends from left to right.
Y-intercept
Every linear equation, whether it ascends or descends, crosses the y-axis at some point. This point is known as the y-intercept and is denoted as \(c\) in the slope-intercept form \(y = mx + c\).
The y-intercept is the value of \(y\) where the line crosses the y-axis (i.e., \(x = 0\)).
In our equation \( y = \frac{2}{3}x - 3 \), the y-intercept is \(-3\).
Additionally, knowing the y-intercept allows for an easy verification of the accuracy of a drawn line on a graph.
The y-intercept is the value of \(y\) where the line crosses the y-axis (i.e., \(x = 0\)).
In our equation \( y = \frac{2}{3}x - 3 \), the y-intercept is \(-3\).
- This means that if you were to graph the line, it would intersect the y-axis at the point \( (0, -3) \).
Additionally, knowing the y-intercept allows for an easy verification of the accuracy of a drawn line on a graph.
Graphing Linear Functions
Graphing linear functions is often one of the first introductions to visualizing mathematical equations and understanding their behavior.
For the equation \( y = \frac{2}{3}x - 3 \), start by plotting the y-intercept, \(-3\), on the y-axis as an anchor point for the line.
After marking the y-intercept:
Each point on this line corresponds to an \((x, y)\) pair that satisfies the equation.
Understanding how to graph such lines aids in visual comprehensions of solutions and equation behavior.
For the equation \( y = \frac{2}{3}x - 3 \), start by plotting the y-intercept, \(-3\), on the y-axis as an anchor point for the line.
After marking the y-intercept:
- Use the slope \( \frac{2}{3} \) to determine other points. Since slope indicates "rise over run," from the y-intercept, move 2 units up (rise) and 3 units to the right (run) to find a new point.
- Draw the line through these points. Extend it in both directions to cover the graph area.
Each point on this line corresponds to an \((x, y)\) pair that satisfies the equation.
Understanding how to graph such lines aids in visual comprehensions of solutions and equation behavior.
Other exercises in this chapter
Problem 35
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