Problem 14 - Precalculus: Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) \(P(4,3), Q(1,-1)\)

Step-by-Step Solution

Answer

The slope of the line is \(\frac{4}{3}\).

Step 1: Identifying the Formula for Slope

The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 2: Substituting Point Coordinates

For points \(P(4, 3)\) and \(Q(1, -1)\), substitute \((x_1, y_1) = (4, 3)\) and \((x_2, y_2) = (1, -1)\) into the slope formula: \[ m = \frac{-1 - 3}{1 - 4} \].

Step 3: Calculating the Numerator

Calculate the difference in the y-coordinates: \(-1 - 3 = -4\).

Step 4: Calculating the Denominator

Calculate the difference in the x-coordinates: \(1 - 4 = -3\).

Step 5: Finding the Slope

Divide the result from the numerator by the denominator to find the slope: \[ m = \frac{-4}{-3} = \frac{4}{3} \].

Key Concepts

Coordinate Geometry

Coordinate geometry is a fascinating area that connects algebra and geometry by using a coordinate system to help describe geometric concepts like lines, curves, and shapes. In coordinate geometry, or analytic geometry, the location of points is defined using coordinates, usually in the form of \((x, y)\). Each point on the plane has a pair of coordinates, where the first number \(x\) is the horizontal distance from the origin, and the second number \(y\) is the vertical distance.
This approach allows us to apply algebraic methods to solve geometric problems. For example, by knowing the coordinates of points \(P(4, 3)\) and \(Q(1, -1)\), we can use mathematical formulas to determine the properties of the line connecting them. Coordinate geometry becomes a very powerful tool, as it helps in analyzing shapes and solving problems that may be difficult to visualize otherwise.

• Understanding how coordinates work is key to describing positions on a plane.
• Coordinate geometry translates geometric problems into algebraic computations.

Slope Formula

The slope formula is a crucial concept used to find the steepness or inclination of a line in coordinate geometry. Slope is often denoted by the letter \(m\) and indicates how much a line rises or falls as it moves from left to right. The slope formula, given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] measures the change in the \( y \)-coordinates relative to the change in the \( x \)-coordinates between two points.
The idea is simple yet profound: slope defines the direction and angle of a line on a plane. In the line through points \(P(4, 3)\) and \(Q(1, -1)\), substituting the corresponding values into the slope formula gives us \[ m = \frac{-1 - 3}{1 - 4} = \frac{-4}{-3} = \frac{4}{3} \]. This result means the line rises 4 units for every 3 units it travels horizontally.Some key points about slope:

• A positive slope means the line inclines upwards.
• A negative slope means the line declines downwards.
• A zero slope means the line is horizontal.

Points on a Plane

Points on a plane are the foundational elements in coordinate geometry. A plane is a flat, two-dimensional surface where every location can be defined using a pair of numbers called coordinates. These numbers are written in the form \((x, y)\):

• The first value \(x\) represents the horizontal axis (or abscissa).
• The second value \(y\) represents the vertical axis (or ordinate).

By plotting points like \(P(4, 3)\) and \(Q(1, -1)\) on the plane, we can visualize linear relationships and apply algebraic processes, such as calculating the slope. Understanding how to represent points and interpret their positions is essential in exploring more complex shapes and their relationships within the plane.
The relationship between multiple points is what leads to shapes like lines and polygons. Knowing the coordinates allows for computations of distances, slopes, and even areas directly on the plane, bridging the gap between pure numbers and geometric intuition.