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Chapter 2: Problem 8

Find the slope of the line through P and Q. $$ P(1,2), Q(3,3) $$

Short Answer

Expert verified
The slope of the line through points P and Q is \( \frac{1}{2} \).

Step by step solution

01

Understand the Slope Formula

To find the slope of the line through two points, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
02

Assign Point Coordinates to Variables

Identify the coordinates of points P and Q. Here, P is \((1, 2)\) which means \(x_1 = 1\) and \(y_1 = 2\). For point Q \((3, 3)\), \(x_2 = 3\) and \(y_2 = 3\).
03

Substitute Values into Slope Formula

Now substitute these values into the slope formula:\[ m = \frac{3 - 2}{3 - 1} \]
04

Simplify the Expression

Calculate the differences: \(y_2 - y_1 = 3 - 2 = 1\) and \(x_2 - x_1 = 3 - 1 = 2\). The slope is then:\[ m = \frac{1}{2} \]

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Key Concepts

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

Finding the Slope
The slope of a line is an important concept in algebra and geometry. It's a measure of the steepness of the line. Understanding how to find it is essential for graphing and interpreting linear relationships.
To find the slope, you'll use the slope formula:
  • \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula helps you calculate the change in the vertical direction (rise) relative to the change in the horizontal direction (run) between two points on the line.
Once you know the positions of two points, you can calculate the slope by substituting their coordinates into the formula, giving you a clear numeric representation of the line's incline.
Coordinates
Coordinates are pairs of numbers that define the position of a point on a plane. In two-dimensional geometry, each point is expressed as
  • o(x, y)
where 'x' represents the horizontal distance and 'y' the vertical distance from a fixed reference point called the origin.
Understanding coordinates is crucial because they provide the necessary information to graph points and lines on a Cartesian plane.
For our task, knowing the coordinates of points P and Q鈥�
  • P(1,2)
  • Q(3,3)
is vital to find the slope accurately. By identifying these coordinates, you lay the groundwork for applying the slope formula effectively.
Mathematics Formula
Mathematical formulas are powerful tools used to simplify complex calculations into manageable steps. The slope formula is a classic example. It condenses the concept of steepness into an easy-to-use equation.
Here's how we apply the slope formula in practice:
  • Identify the coordinates from the points, P(1,2) and Q(3,3).
  • Plug these coordinates into the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  • Calculate: \[ m = \frac{3 - 2}{3 - 1} = \frac{1}{2} \]
By following these steps, the daunting task of determining the line's slope becomes straightforward.
Mathematical formulas not only streamline difficult concepts but also make calculations efficient and error-free.

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Most popular questions from this chapter

Drug Dosages If the recommended adult dosage for a drug is \(D\) (in mg), then to determine the appropriate dosage \(c\) for a child of age \(a\) , pharmacists use the equation $$ c=0.0417 D(a+1) $$ Suppose the dosage for an adult is 200 \(\mathrm{mg}\) (a) Find the slope. What does it represent? (b) What is the dosage for a newborn?

\(55-58\) m Find all real solutions of the equation, rounded to two decimals. $$ x(x-1)(x+2)=\frac{1}{6} x $$

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ \sqrt{0.5 x^{2}+1} \leq 2|x| $$

Crickets and Temperature Biologists have observed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 120 chirps per minute at \(70^{\circ} \mathrm{F}\) and 168 chirps per minute at \(80^{\circ} \mathrm{F} .\) (a) Find the linear equation that relates the temperature \(t\) and the number of chirps per minute \(n\) . (b) If the crickets are chirping at 150 chirps per minute, estimate the temperature.

Find an equation of the line that satisfies the given conditions. Through \((-3,-5) ;\) slope \(-\frac{7}{2}\)
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