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Chapter 3: Problem 54

Identify the slope of a line that is: \- parallel to the line \(y=3 x-14\).

Short Answer

Expert verified
The slope is 3.

Step by step solution

01

Identify the form of the given line equation

Recognize that the given equation is in the slope-intercept form, which is \( y = mx + b \). Here, \( m \) represents the slope and \( b \) is the y-intercept.
02

Determine the slope of the given line

In the equation \( y = 3x - 14 \), the coefficient of \( x \) is 3. Therefore, the slope \( m \) of the given line is 3.
03

Find the slope of the parallel line

Lines that are parallel have the same slope. Since the slope of the given line is 3, the slope of a line parallel to it will also be 3.

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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 lines that never meet, no matter how far they are extended. Think of railroad tracks. They always run side by side. The key feature of parallel lines is that they have the same slope. If you have two lines with the same slope, they will never cross each other.
For instance, if we have a line with the equation: \( y = 3x - 14 \), the slope is 3.
Any line that is parallel to this one will also have a slope of 3.
This concept is very useful when you want to identify whether two lines are parallel, just compare their slopes!
slope-intercept form
The slope-intercept form of a line's equation is one of the most common ways to express a linear equation. It is written as:
\(y = mx + b\)
* \( m \) is the slope of the line.
* \( b \) is the y-intercept, or the point where the line crosses the y-axis.
In this form, it鈥檚 easy to see the slope right away. For example, in the equation:
\( y = 3x - 14 \), the slope \( m = 3 \) and the y-intercept \( b = -14 \). Using the slope-intercept form, you can quickly sketch a graph of the line and identify slopes of parallel lines.
y-intercept
The y-intercept is a key point in understanding a linear equation. It is the place where the line crosses the y-axis.
In the slope-intercept form: \( y = mx + b \), the y-intercept is represented by \( b \).
To find the y-intercept, set \( x \) to 0 in the equation. For example, in \( y = 3x - 14 \):
If \( x = 0 \), then \( y = -14 \).
So, the y-intercept of this line is -14.
This point gives you a starting place to draw the line on a graph.

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Most popular questions from this chapter

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((1,4) ;(3,-6)\)

(a) use a graphing calculator to graph \(y=\frac{3}{4} x-5\). Sketch the graph; describe the window. (b) for the given value of \(x\), use the Value command to find the value of \(y\). If it is necessary to change the window, describe the changed window. \(x=20\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{6}, 8\right) ;\left(\frac{5}{6}, 11\right)\)

(a) write the equation of the vertical line that passes through the point. (b) graph the equation. \((-2,-1)\)

(a) find three solutions of the equation. (b) graph the equation. \(y=-\frac{1}{4} x+3\)
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