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Chapter 2: Problem 23

Find the slope of the line passing through the following pair of points. (3,4) and (3,7)

Short Answer

Expert verified
The slope of the line passing through the points (3,4) and (3,7) is undefined. Because the line is vertical.

Step by step solution

01

Identify the coordinates

Identify the coordinates of the points. As per the problem, point 1 is (3,4) which makes x1=3 and y1=4. Point 2 is (3,7), this makes x2=3 and y2=7.
02

Substitute the coordinates in the slope formula

Substitute the values into the slope formula \( m = \frac{(y2 - y1)}{(x2 - x1)} \). Now, \( m = \frac{(7 - 4)}{(3 - 3)} \).
03

Solve for m

Solve the equation for m. The equation thus becomes \( m = \frac{3}{0} \). However, division by 0 is undefined in mathematics, so m is undefined. This means that the line passing through the points (3,4) and (3,7) is vertical, so its slope is undefined.

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Key Concepts

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

Undefined Slope
When we talk about the 'slope' of a line, we think of how steep the line is, often represented as "rise over run." Mathematically, this is shown using the formula:
  • \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \)
In cases where the denominator (\(x_2 - x_1\)) equals zero, you cannot divide by zero. Thus, the slope is undefined.

Why can't we divide by zero? It's because division by zero doesn't make sense in mathematics, as it doesn't give us a real or finite number.

Hence, in our exercise, since the x-coordinates of the points (3,4) and (3,7) are the same, the slope \( m = \frac{(7 - 4)}{(3 - 3)} \) becomes \( \frac{3}{0} \), meaning the line is vertical, and the slope is undefined.
Coordinate Geometry
Coordinate Geometry is the branch of mathematics that uses coordinates to describe and analyze geometric properties.

It uses an ordered pair of numbers, known as coordinates, to locate points on a plane, typically referred to as the Cartesian plane.
For instance:
  • First value in a pair (x-coordinate) indicates the position along the horizontal x-axis.
  • Second value (y-coordinate) defines the position on the vertical y-axis.
In our given problem, the points (3,4) and (3,7) are located on the Cartesian plane. By comparing their x and y coordinates, we're able to discuss attributes of the line connecting these points, such as slope.

Using these coordinates allows us a clear system to understand different aspects of geometry and algebra through a visual lens.
Vertical Line
A vertical line is a straight path that goes up and down perfectly on a graph.

What makes it special? A vertical line has an undefined slope and is represented by an equation where \(x = c\), where \(c\) is a constant.
  • It doesn't lean left or right, just straight up and down.
  • All points on a vertical line have the same x-coordinate.
In our example, the points (3,4) and (3,7) form a vertical line because both share the same x-coordinate of 3. So the equation of the line is \(x = 3\).

Since a vertical line doesn't 'rise or run,' the slope is undefined. This concept is fundamental as it distinguishes vertical lines from all other line types on a graph.

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Most popular questions from this chapter

The variable cost to manufacture an item is $$\$ 10,$$ and it costs $$\$ 2,500$$ to produce 100 items. Write the cost function, and use this function to estimate the cost of manufacturing 300 items.

Find the slope of the line passing through the following pair of points. (6,-5) and (4,-1)

Assume a linear relationship holds. To manufacture 30 items, it costs $$\$ 2700$$, and to manufacture 50 items, it costs $$\$ 3200$$. If \(x\) represents the number of items manufactured and \(y\) the cost, write the cost function.

Assume a linear relationship holds. The freezing temperatures for Celsius and Fahrenheit scales are 0 degree and 32 degrees, respectively. The boiling temperatures for Celsius and Fahrenheit are 100 degrees and 212 degrees, respectively. Let \(C\) denote the temperature in Celsius and \(F\) in Fahrenheit. Write the conversion function from Celsius to Fahrenheit, and use this function to convert 25 degrees Celsius into an equivalent Fahrenheit measure.

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It has \(x\) -intercept \(=3\) and \(y\) -intercept \(=4\).
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