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Chapter 3: Problem 13

Determine the slope of parallel lines and perpendicular lines. $$ 4 x+3 y=0 $$

Short Answer

Expert verified
The slope of parallel lines is \(-\frac{4}{3}\), and the slope of perpendicular lines is \(\frac{3}{4}\).

Step by step solution

01

Rewrite the equation in slope-intercept form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope. Start by rewriting the given equation \(4x + 3y = 0\) into this form. To do this, solve for \(y\) in terms of \(x\). First, subtract \(4x\) from both sides: \[3y = -4x\]Then divide both sides by 3 to isolate \(y\): \[y = -\frac{4}{3}x\] Now, the equation is in slope-intercept form, and the slope \(m\) is \(-\frac{4}{3}\).
02

Find the slope of a parallel line

Lines that are parallel have the same slope. Therefore, if a line is parallel to the given line \(4x + 3y = 0\), it will also have a slope of \(-\frac{4}{3}\).
03

Find the slope of a perpendicular line

Lines that are perpendicular have slopes that are negative reciprocals of each other. Since the slope of the given line is \(-\frac{4}{3}\), the slope of a line perpendicular to it would be the negative reciprocal of \(-\frac{4}{3}\), which is \(\frac{3}{4}\). To find the negative reciprocal, flip the fraction to \(\frac{3}{4}\) and change the sign from negative to positive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallel Lines
Parallel lines are fascinating in geometry because they never intersect, no matter how far they are extended.
This unique property is due to them having equal slopes. When two lines are parallel, they run in the same direction.
This means the steepness or slope of the lines is identical.
  • The slope of a line is represented by the letter "m" in the slope-intercept equation, which is \( y = mx + b \).
  • For the equation \( 4x + 3y = 0 \), the slope is \(-\frac{4}{3}\) after converting this into the slope-intercept form: \( y = -\frac{4}{3}x \).
  • So, any line parallel to \( 4x + 3y = 0 \) will also have a slope of \(-\frac{4}{3}\).
This consistency ensures that parallel lines, which play a crucial role in coordinate geometry, stay forever apart, thus maintaining a constant gap throughout their entire lengths.
Perpendicular Lines
Perpendicular lines, unlike parallel lines, intersect each other at right angles, which is a 90-degree angle.
This is why they are perfect for creating grids and architectural designs that require strong intersections. To determine the slope of a line perpendicular to another, we need to look at the slopes themselves.
The slopes of perpendicular lines are always negative reciprocals of each other.
  • The slope of the original line from the equation \( 4x + 3y = 0 \) is \(-\frac{4}{3}\).
  • The negative reciprocal of this slope means flipping the fraction and changing the sign:
    it becomes \(\frac{3}{4}\).
  • This new slope, \(\frac{3}{4}\), guarantees that a line with this slope will meet the original line at a 90-degree angle.
This geometric relationship ensures that perpendicular lines create precise, controlled intersections, perfect for structural designs and more in geometry.
Slope-Intercept Form
The slope-intercept form of a line is a crucial tool in algebra and geometry, making it easy to understand both the slope and position of a line graphically. This equation is written as \(y = mx + b\), where:
  • "m" represents the slope, indicating how steep the line is.
  • "b" is the y-intercept, showing where the line crosses the y-axis.
When converting equations to this form, as we did with \(4x + 3y = 0\), it makes the slope immediately visible,
in this case, \(-\frac{4}{3}\), and helps graph the line accurately.Besides finding slopes, the slope-intercept form tells us about the line鈥檚 initial value on the y-axis,
helping us understand its behavior in different scenarios. By mastering slope-intercept form, solving many algebraic and geometric problems becomes much more straightforward.

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