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Chapter 10: Problem 23

In Exercises 21 to \(24,\) state whether the lines are parallel, perpendicular, the same (coincident), or none of these. $$2 x+3 y=6 \text { and } 3 x-2 y=12$$

Short Answer

Expert verified
The lines are perpendicular.

Step by step solution

01

Convert to Slope-Intercept Form

Start by converting each line equation into the slope-intercept form \( y = mx + b \). For the first equation, \( 2x + 3y = 6 \), solve for \( y \):\[ 3y = -2x + 6 \]Divide each term by 3:\[ y = -\frac{2}{3}x + 2 \]Thus, the slope \( m_1 \) of the first line is \( -\frac{2}{3} \). Next, for the second equation \( 3x - 2y = 12 \), solve for \( y \):\[ -2y = -3x + 12 \]Divide each term by -2:\[ y = \frac{3}{2}x - 6 \]Thus, the slope \( m_2 \) of the second line is \( \frac{3}{2} \).
02

Compare the Slopes

Now that you have both slopes, compare them to determine the relationship between the lines:1. **Parallel Lines**: Slopes are equal.2. **Perpendicular Lines**: The product of the slopes is -1.3. **Coincident Lines**: Lines have the same equation.The slopes are \( m_1 = -\frac{2}{3} \) and \( m_2 = \frac{3}{2} \). Calculate the product of the slopes:\[ m_1 \times m_2 = -\frac{2}{3} \times \frac{3}{2} = -1 \]Since the product of the slopes 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
Slope-intercept form is a way to write the equation of a line so it is easy to understand its properties, specifically the slope and the y-intercept. The form is given by:
  • \(y = mx + b\)
Here, \(m\) is the slope, which tells us how steep the line is, and \(b\) is the y-intercept, which is where the line crosses the y-axis. The slope gives you the vertical change for every horizontal change of 1 unit.
To find the slope-intercept form from the standard form of an equation like \(ax + by = c\), solve for \(y\):
  • First, isolate \(y\) by moving \(x\)-terms to the other side.
  • Then, divide all terms by \(b\) to get \(y\) by itself.
This conversion allows you to clearly see the slope and y-intercept, making it easier to analyze and graph the line.
Parallel Lines
Parallel lines are two or more lines in a plane that never intersect. They remain the same distance apart over their entire length. For lines to be parallel:
  • The slopes of the lines must be equal. They have to rise over the same run.
Let's say you have two lines in the form \(y = m_1x + b_1\) and \(y = m_2x + b_2\). If \(m_1 = m_2\), the lines are parallel. However, they will not coincide unless \(b_1 = b_2\) as well.
This equality in slopes ensures that both lines move in exactly the same direction, never meeting at any point. A real-world example is train tracks, where both tracks are parallel to ensure a train rides safely along them.
Coincident Lines
Coincident lines are essentially the same line, meaning they overlap completely. This happens when two lines have:
  • The same slope, hence the same direction.
  • The same y-intercept, meaning they start from the same point on the y-axis.
To determine if lines are coincident, compare their equations. If their line equations are identical, then they lie atop one another. For instance, lines described by equations \(y = -\frac{2}{3}x + 2\) and \(y = -\frac{2}{3}x + 2\) are coincident because both equations match exactly. Coincident lines are rare if we are just making up equations randomly, but they are a perfect match by definition.

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Most popular questions from this chapter

Apply the Midpoint Formula. \(M(3,-4)\) is the midpoint of \(\overline{A B},\) in which \(A\) is the point \((-5,7) .\) Find the coordinates of \(B\)

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For isosceles \(\triangle P N Q,\) the vertices are \(P(-2 a, 0), N(2 a, 0)\) and \(Q(0,2 b) .\) In terms of \(a\) and \(b,\) find the coordinates of the circumcenter of \(\triangle P N Q .\) (The circumcenter is the point of concurrence for the perpendicular bisectors of the sides of a triangle.)
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