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Magazine Chapter 8: Problem 11
For Problems 1-36, graph each linear equation. (Objective 2) $$ 3 x-2 y=6 $$
Short Answer
Expert verified
The line crosses the y-axis at -3, has a slope of \(\frac{3}{2}\), and another point on the line is \((2, 0)\).
Step by step solution
01
Rearrange Equation To Slope-Intercept Form
To graph a linear equation, it's often easiest to use the slope-intercept form, which is given by \( y = mx + b \). Rearrange the equation \( 3x - 2y = 6 \) to solve for \( y \). First, move \( 3x \) to the right side: \( -2y = -3x + 6 \). Then, divide all terms by \(-2\) to solve for \( y \): \( y = \frac{3}{2}x - 3 \).
02
Identify the Slope and y-Intercept
The rearranged equation \( y = \frac{3}{2}x - 3 \) is now in the form \( y = mx + b \). Here, \( m = \frac{3}{2} \) represents the slope, and \( b = -3 \) represents the y-intercept. This means the graph crosses the y-axis at \( y = -3 \).
03
Plot the y-Intercept
Plot the point \((0, -3)\) on the graph. This is the y-intercept, where the line crosses the y-axis.
04
Use the Slope to Find Another Point
The slope \( \frac{3}{2} \) means that for every 2 units you move to the right (along the x-axis), you move up 3 units (along the y-axis). Starting from the y-intercept \((0, -3)\), move 2 units to the right to \((2, -3)\), and then 3 units up to \((2, 0)\). Plot this second point \((2, 0)\) on the graph.
05
Draw the Line
Draw a straight line through the points \((0, -3)\) and \((2, 0)\). Extend the line across the graph to represent all solutions to the equation \(3x - 2y = 6\).
06
Verify Additional Points
To ensure accuracy, you can test another x-value, such as \( x = 4 \). Substitute \( x = 4 \) into the equation \( 3x - 2y = 6 \): \( 3(4) - 2y = 6 \) which simplifies to \( 12 - 2y = 6 \). Solving for \( y \) gives \( y = 3 \). Thus, the point \((4, 3)\) should lie on the line. Confirm this by checking the graph.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate plane and drawing a line through these points. This process visually represents the solutions of a linear equation. Each equation of a line in the form of \( Ax + By = C \) can be transformed into simpler formats for graphing.
To begin, you can use the slope-intercept form, where finding particular points such as the y-intercept is key. Another point can be found using the slope. Once two or more points are plotted, a straight line connects these points, illustrating the equation's solution.
To begin, you can use the slope-intercept form, where finding particular points such as the y-intercept is key. Another point can be found using the slope. Once two or more points are plotted, a straight line connects these points, illustrating the equation's solution.
- Choose key points from the equation or calculated points that satisfy the equation.
- Plot these points on the specified graph axes.
- Draw a line through all plotted points.
Slope-Intercept Form
The slope-intercept form is an easy-to-use format for graphing linear equations. It is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) represents where the line crosses the y-axis.
This form is particularly beneficial because it straightforwardly provides two critical elements: the slope and the y-intercept. To transition an equation into this form, you often need to solve for \( y \), rearranging the terms accordingly.
This form is particularly beneficial because it straightforwardly provides two critical elements: the slope and the y-intercept. To transition an equation into this form, you often need to solve for \( y \), rearranging the terms accordingly.
- Solve for \( y \) if needed, expressing the equation in the format \( y = mx + b \).
- Identify both the slope and y-intercept from this arrangement for plotting.
Y-Intercept
The y-intercept is a crucial point on the graph where the line intersects the y-axis. It is the value of \( y \) when \( x = 0 \).
In the slope-intercept form \( y = mx + b \), the constant \( b \) denotes the y-intercept. For graphing, this point is often plotted first as it provides a starting anchor for the line.
In the slope-intercept form \( y = mx + b \), the constant \( b \) denotes the y-intercept. For graphing, this point is often plotted first as it provides a starting anchor for the line.
- Identify \( b \) from the slope-intercept equation.
- Plot the point on the y-axis at \( (0, b) \).
Slope in Linear Equations
The slope of a linear equation is a measure of how steep the line is. It is the rate of change of \( y \) with respect to \( x \). Slope can tell you how fast a line ascends or descends
In the slope-intercept form \( y = mx + b \), "\( m \)" stands for the slope. A positive slope moves upward from left to right, whereas a negative slope moves downward. To use the slope for graphing:
In the slope-intercept form \( y = mx + b \), "\( m \)" stands for the slope. A positive slope moves upward from left to right, whereas a negative slope moves downward. To use the slope for graphing:
- Determine the slope \( m \).
- From the y-intercept \((0, b)\), apply \( m \) as a ratio of rise over run.
- Moving horizontally (the run), then vertically (the rise), plot additional points.
