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Magazine Chapter 4: Problem 374
Use slopes and y-intercepts to determine if the lines are perpendicular. \(3 x-2 y=1 ; 2 x-3 y=2\)
Short Answer
Expert verified
The lines are not perpendicular because the slopes \( \frac{3}{2} \) and \( \frac{2}{3} \) are not negative reciprocals.
Step by step solution
01
Write Equations in Slope-Intercept Form
Convert the given equations to the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.Starting with the first equation: \(3x - 2y = 1\)Solve for \(y\): \(3x - 1 = 2y\) \(y = \frac{3}{2}x - \frac{1}{2}\)
02
Find the Slope of the First Line
From the slope-intercept form \(y = \frac{3}{2}x - \frac{1}{2}\), the slope \(m_1\) of the first line is \(\frac{3}{2}\).
03
Write the Second Equation in Slope-Intercept Form
Convert the second equation to slope-intercept form:\(2x - 3y = 2\)Solve for \(y\): \(2x - 2 = 3y\) \(y = \frac{2}{3}x - \frac{2}{3}\)
04
Find the Slope of the Second Line
From the slope-intercept form \(y = \frac{2}{3}x - \frac{2}{3}\), the slope \(m_2\) of the second line is \(\frac{2}{3}\).
05
Determine if the Slopes are Negative Reciprocals
To check if the lines are perpendicular, determine if the slopes \(m_1\) and \(m_2\) are negative reciprocals. Negative reciprocals of each other multiply to \(-1\): \(\frac{3}{2} \cdot \frac{2}{3} = 1\,eq\,-1\)Since the product of the slopes is not \(-1\), the lines are not perpendicular.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope-intercept form
To analyze lines, we often use the slope-intercept form of a linear equation. This form is represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
The slope \( m \) indicates the steepness of the line, and the y-intercept \( b \) indicates where the line crosses the y-axis.
Converting an equation into this form helps to easily identify these characteristics.
For example, let's convert the equation \( 3x - 2y = 1 \) to slope-intercept form:
The slope \( m \) indicates the steepness of the line, and the y-intercept \( b \) indicates where the line crosses the y-axis.
Converting an equation into this form helps to easily identify these characteristics.
For example, let's convert the equation \( 3x - 2y = 1 \) to slope-intercept form:
- Start by solving for \( y \).
- Rearrange to isolate \( y \) on one side: \, \( 3x - 1 = 2y \).
- Divide through by -2: \, \( y = \frac{3}{2}x - \frac{1}{2} \).
negative reciprocals
Perpendicular lines have a special relationship in terms of their slopes. The slopes of perpendicular lines are negative reciprocals of each other.
This means that if one line has a slope \( m \), the other line's slope will be \( -\frac{1}{m} \).
For example, if a line has a slope of \( \frac{3}{2} \), then a line perpendicular to it will have a slope of \( -\frac{2}{3} \).
To determine if two lines are perpendicular, simply multiply their slopes and see if the product is \( -1 \).
Example:
This means that if one line has a slope \( m \), the other line's slope will be \( -\frac{1}{m} \).
For example, if a line has a slope of \( \frac{3}{2} \), then a line perpendicular to it will have a slope of \( -\frac{2}{3} \).
To determine if two lines are perpendicular, simply multiply their slopes and see if the product is \( -1 \).
Example:
- Line 1 slope: \( \frac{3}{2} \).
- Line 2 slope: \( \frac{2}{3} \).
- Multiplied together: \( \frac{3}{2} \cdot \frac{2}{3} = 1 \), which does not equal \( -1 \).
slope calculation
Calculating the slope of a line helps in understanding its direction and steepness. Slope can be derived from an equation or using two points on the line.
From an equation in slope-intercept form \( y = mx + b \), the slope is directly given by \( m \).
For instance, for \( y = \frac{3}{2}x - \frac{1}{2} \), the slope is \( \frac{3}{2} \).
To calculate slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Example:
From an equation in slope-intercept form \( y = mx + b \), the slope is directly given by \( m \).
For instance, for \( y = \frac{3}{2}x - \frac{1}{2} \), the slope is \( \frac{3}{2} \).
To calculate slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Example:
- Points: (1, 2) and (3, 8).
- Calculate differences: \( 8 - 2 \) and \( 3 - 1 \).
- Apply formula: \( m = \frac{8-2}{3-1} = \frac{6}{2} = 3 \).
equation conversion
Equation conversion is crucial to understanding the properties of a line. Let鈥檚 convert standard form equations to slope-intercept form.
Example equation: \( 3x - 2y = 1 \).
Steps to convert:
For another example, let鈥檚 convert \( 2x - 3y = 2 \):
Example equation: \( 3x - 2y = 1 \).
Steps to convert:
- Isolate the y-term: \( 3x - 1 = 2y \).
- Divide both sides by the coefficient of y: \( y = \frac{3}{2}x - \frac{1}{2} \).
For another example, let鈥檚 convert \( 2x - 3y = 2 \):
- Isolate the y-term: \( 2x - 2 = 3y \).
- Divide by 3: \( y = \frac{2}{3}x - \frac{2}{3} \).
