Problem 373 Use slopes and y-intercepts to d... [FREE SOLUTION]

Chapter 4: Problem 373

Use slopes and y-intercepts to determine if the lines are perpendicular. \(2 x+3 y=5 ; 3 x-2 y=7\)

Short Answer

Expert verified

The lines are perpendicular because the slopes are negative reciprocals (\(m_1 = -\frac{2}{3}\) and \(m_2 = \frac{3}{2}\)).

Step by step solution

Convert to Slope-Intercept Form

To determine if two lines are perpendicular, convert each equation to the slope-intercept form, which is \(y = mx + b\). This form makes it easy to identify the slope (\(m\)).For the first equation, 2x + 3y = 5:Subtract 2x from both sides:3y = -2x + 5Divide every term by 3:y = -\(\frac{2}{3}\)x + \(\frac{5}{3}\)Here, the slope (\(m_1\)) is -\(\frac{2}{3}\).For the second equation, 3x - 2y = 7:Subtract 3x from both sides:-2y = -3x + 7Divide every term by -2:y = \(\frac{3}{2}\)x - \(\frac{7}{2}\)Here, the slope (\(m_2\)) is \(\frac{3}{2}\).

Determine If Slopes Are Negative Reciprocals

Lines are perpendicular if the product of their slopes is -1. Check if the slopes \(m_1\) and \(m_2\) are negative reciprocals of each other:a) Identify the slopes from Step 1:\(m_1 = -\frac{2}{3}\) and \(m_2 = \frac{3}{2}\)b) Calculate their product:\(m_1 \cdot m_2 = -\frac{2}{3} \cdot \frac{3}{2} = -1\)Since the product is -1, the lines are perpendicular.

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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 expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Converting any linear equation into this form makes it easy to identify the slope and y-intercept.

Negative Reciprocals

Two numbers are negative reciprocals if their product is -1. For lines, their slopes are negative reciprocals if the lines are perpendicular. In this case, if \(m_1\) is the slope of one line and \(m_2\) is the slope of another line, \(m_1 \times m_2 = -1\) implies that the lines are perpendicular.

Linear Equations

A linear equation represents a straight line in the form \(ax + by = c\). Each linear equation can be transformed into the slope-intercept form, making it easier to analyze and determine relationships between lines, such as parallelism or perpendicularity.

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91影视

Use slopes and y-intercepts to determine if the lines are perpendicular. \(2 x+3 y=5 ; 3 x-2 y=7\)

Short Answer

Step by step solution

Convert to Slope-Intercept Form

Determine If Slopes Are Negative Reciprocals

Key Concepts

Slope-Intercept Form

Negative Reciprocals

Linear Equations

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