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Magazine Chapter 3: Problem 115
In the following exercises, graph the line of each equation using its slope and \(y\) -intercept. $$ 3 x-2 y=4 $$
Short Answer
Expert verified
Slope = \( \frac{3}{2} \), y-intercept = -2.
Step by step solution
01
- Rewrite the equation in slope-intercept form
First, rewrite the given equation in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start with the given equation: \[ 3x - 2y = 4 \] Solve for \( y \) by isolating it on one side of the equation.
02
- Isolate the y variable
Add \( 2y \) to both sides to move it to the right side of the equation: \[ 3x = 2y + 4 \].Next, subtract 4 from both sides: \[ 3x - 4 = 2y \] Now, divide every term by 2 to solve for \( y \): \[ y = \frac{3}{2}x - 2 \]
03
- Identify the slope and y-intercept
In the equation \( y = \frac{3}{2}x - 2 \), the slope (\( m \)) is \( \frac{3}{2} \) and the y-intercept (\( b \)) is -2. This means the line crosses the y-axis at -2 and rises 3 units for every 2 units it moves to the right.
04
- Plot the y-intercept
Begin by plotting the y-intercept, which is the point (0, -2) on the coordinate plane.
05
- Use the slope to plot another point
From the y-intercept (0, -2), use the slope \( \frac{3}{2} \). Move up 3 units and to the right 2 units to find the next point. This will put you at (2, 1).
06
- Draw the line
Using the points (0, -2) and (2, 1), draw a straight line through these points to represent the equation.
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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 linear equations easy, we often use the slope-intercept form of a line's equation. This form is represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. By converting an equation to this form, we can quickly identify key characteristics of the line.
For example, let's consider the equation \( 3x - 2y = 4 \). To change this into slope-intercept form, solve for \( y \).
For example, let's consider the equation \( 3x - 2y = 4 \). To change this into slope-intercept form, solve for \( y \).
- First, add 2y to move it to the other side: \( 3x = 2y + 4 \).
- Then, subtract 4 from both sides: \( 3x - 4 = 2y \).
- Lastly, divide the entire equation by 2: \( y = \frac{3}{2}x - 2 \).
slope
The slope \( (m) \) in the equation of a line tells us how steep the line is. It shows the rate of change of \( y \) with respect to \( x \). The slope is usually written as a fraction \( \frac{rise}{run} \), where 'rise' is how much we move up or down and 'run' is how much we move left or right.
In our example equation \( y = \frac{3}{2}x - 2 \), the slope \( m \) is \( \frac{3}{2} \). This means for every 2 units we move to the right (run), the line goes up by 3 units (rise).
To graph:
In our example equation \( y = \frac{3}{2}x - 2 \), the slope \( m \) is \( \frac{3}{2} \). This means for every 2 units we move to the right (run), the line goes up by 3 units (rise).
To graph:
- Start from any point (usually from the y-intercept).
- Then use the slope to find another point.
- In this example, from (0, -2), go up 3 units and right 2 units to mark the next point at (2, 1).
y-intercept
The y-intercept \( (b) \) is the point where the line crosses the y-axis. It tells us the value of \( y \) when \( x = 0 \).
In our example, the y-intercept is -2. This means the line will cross the y-axis at the point (0, -2).
To locate the y-intercept on the graph:
In our example, the y-intercept is -2. This means the line will cross the y-axis at the point (0, -2).
To locate the y-intercept on the graph:
- Find the vertical axis (y-axis) on the coordinate plane.
- Move up or down to the value of \( b \).
- Here, plot a point at (0, -2).
coordinate plane
A coordinate plane is a two-dimensional surface where we can graph equations and visualize their solutions. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. The intersection of these axes at (0, 0) is called the origin.
To graph our equation \( y = \frac{3}{2}x - 2 \) on the coordinate plane:
To graph our equation \( y = \frac{3}{2}x - 2 \) on the coordinate plane:
- First, plot the y-intercept point at (0, -2).
- Using the slope \( \frac{3}{2} \), find another point from the y-intercept by moving up 3 units and right 2 units to get (2, 1).
- Draw a straight line through these points.
