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Chapter 3: Problem 48

Find the slope of the line determined by each equation. passing through (3,-5) and (3,14)

Short Answer

Expert verified
The slope of the line passing through points (3,-5) and (3,14) is undefined because the line is vertical.

Step by step solution

01

Identify the coordinates

First, clearly identify the coordinates of the two points. Point 1 is (3, -5) which gives \( x1 = 3 \) and \( y1 = -5 \). Point 2 is (3,14) which gives \( x2 = 3 \) and \( y2 = 14 \).
02

Substitute into the formula

Substitute the identified coordinates into the slope formula. This will give : \( m = \frac{(14 - (-5))}{(3-3)} \) which simplifies to \( m = \frac{19}{0} \) here something interesting happens, a division by zero occurs.
03

Handle division by zero

By mathematical convention, division by zero is undefined, this means that the slope of the line is also undefined. This happens because the line is vertical.

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Key Concepts

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

Slope Formula
The slope of a line is a measure of how steep the line is. It tells you how much the line goes up or down as you move from left to right. The slope is calculated using the slope formula, which requires two points on a line.

The formula for calculating the slope, often represented as \( m \), is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points on the line. Subtract the \(y\)-coordinates to find the "rise," and subtract the \(x\)-coordinates to find the "run." The slope, \( m \), is the ratio of these two differences.
Vertical Line
A vertical line is a special type of line. It goes straight up and down, unlike other lines that might slant. When you have a vertical line, all the points on this line have the same \( x \)-coordinate.

In the case of a vertical line, the slope formula results in a division by zero. This is because the "run," or the difference in the \( x \)-coordinates, is zero. This makes vertical lines unique because they do not have a defined slope, since you cannot divide by zero.

So, when you see a vertical line, remember:
  • All \( x \)-coordinates are identical.
  • The slope is undefined because the line does not tilt to the left or right.
Division by Zero
Division by zero is an important concept in mathematics, mainly because it is undefined. Imagine trying to split a number into zero parts鈥攊t doesn't make sense, right?

In the slope formula, if the denominator (bottom part of the fraction) is zero, then you've encountered this situation. This is exactly what happens with vertical lines. Since the change in the \( x \)-coordinates (run) is zero, you can't calculate a numerical value for the slope.

Keep in mind when dealing with division by zero:
  • Look for reasons any math problem leads to zero in the denominator.
  • Understand that when it happens, the result is undefined.
Coordinates
Coordinates are a way to represent points on a plane. They come in pairs and are written as \((x, y)\). Each pair tells you exactly where a point is on the Cartesian coordinate system.

The \( x \)-coordinate tells you how far to go left or right, and the \( y \)-coordinate tells you how far to go up or down. By using coordinates, you can easily identify the exact position of a point on a graph.

When working with problems like finding the slope:
  • Read and identify the coordinates correctly.
  • Remember that each point gets an \( x \)- and a \( y \)-coordinate.
Understanding coordinates helps you plug values accurately into the slope formula.

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Most popular questions from this chapter

Set up a variation equation and solve for the requested value. For a fixed area, the length of a rectangle is inversely proportional to its width. A rectangle has a width of 8 feet and a length of 10 feet. If the length is increased to 16 feet, find the width of the rectangle.

Express each combined variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 4. (OBJECTIVE 4) \(y\) varies directly with \(a\) and inversely with \(b .\) If \(y=3\) when \(a=4\) and \(b=12,\) find \(y\) when \(a=10\) and \(b=18\).

Graph each equation. SEE EXAMPLE 6. (OBJECTIVE 5 ) $$y=4$$

Explain the process used to find the \(x\)- and \(y\)-intercepts of the graph of a line.

Solve each inequality and graph the solution. $$-5 x+4 \geq-6$$
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