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Chapter 3: Problem 93

Find the slope of a line parallel to the line \(f(x)=-\frac{7}{2} x-6\)

Short Answer

Expert verified
The slope of the parallel line is \(-\frac{7}{2}\).

Step by step solution

01

Identify the Slope of the Given Line

For a line expressed in the format \( f(x) = mx + b \), the number \( m \) represents the slope. In the equation \( f(x) = -\frac{7}{2}x - 6 \), the slope \( m \) is \(-\frac{7}{2}\).
02

Understanding Parallel Lines

Parallel lines have identical slopes. Therefore, the slope of any line parallel to another is the same as the slope of the given line.
03

Determine the Slope of the Parallel Line

Since a line parallel to the given line \( f(x) = -\frac{7}{2}x - 6 \) must have the same slope, the slope of the desired parallel line is also \(-\frac{7}{2}\).

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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 as they never intersect and maintain a constant distance from each other.
They are like siblings who keep the same pace heading in the same direction.
In terms of linear equations, two lines are parallel if and only if they have the same slope. Here's what you need to remember:
  • If two lines have the same slope, they are parallel.
  • Parallel lines on a graph will never meet, meaning they have no points in common aside from infinity.
For example, if you have the line with the equation \( f(x) = -\frac{7}{2}x - 6 \), any line with the slope \( -\frac{7}{2} \) will be parallel to it no matter its y-intercept.
Linear Equations
Linear equations are straightforward as they represent lines on a coordinate plane.
They are those equations that, when plotted, give you a straight line. A linear equation typically takes the form of \( y = mx + b \), where:
  • \( m \) is the slope of the line, indicating how steep the line is.
  • \( b \) is the y-intercept, which shows where the line crosses the y-axis.
Understanding linear equations is all about recognizing the slope and intercept. The slope tells you if the line goes uphill or downhill, and how quickly.
The y-intercept number tells you the starting point of the line when \( x = 0 \). For the equation \( f(x) = -\frac{7}{2}x - 6 \), the slope \( m \) is \(-\frac{7}{2}\) and the y-intercept \( b \) is -6.
Equation of a Line
The equation of a line is a mathematical way to describe a line on a graph.
It's like a recipe that tells you how to plot a line, using ingredients like slope and intercepts. A standard form of this is the slope-intercept form: \( y = mx + b \). Here's what it includes:
  • **Slope \( m \):** Determines the line's direction and steepness.
  • **Y-Intercept \( b \):** Where the line crosses the y-axis, essentially where it starts when \( x \) is zero.
To find a particular line's equation, you can use two known points or a point and the slope.
For parallel lines, remember they must share the same slope. For example, if you know your line has a slope of \(-\frac{7}{2}\), you should look for lines with the same slope if they need to be parallel.

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Most popular questions from this chapter

Without graphing, find the domain of each function. $$ g(x)=-3 \sqrt{x+5} $$

From the Chapter 3 opener, we have two functions to describe the percent of college students taking at least one online course. For both functions, \(x\) is the number of years since 2000 and \(y\) (or \(f(x)\) or \(g(x))\) is the percent of students taking at least one online course. $$f(x)=2.7 x+4.1 \text { or } g(x)=0.07 x^{2}+1.9 x+5.9$$ Use Exercises \(81-84\) and compare \(f(9)\) and \(g(9),\) then \(f(16)\) and \(g(16) .\) As \(x\) increases, are the function values staying about the same or not? Explain your answer.

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