Problem 12 - Precalculus: Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) \(P(-5,1), Q(3,-2)\)

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{3}{8}\).

Step 1: Understand the Slope Formula

The slope of a line is given by 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 two points on the line. Here, \(P\) corresponds to \((x_1, y_1)\) and \(Q\) to \((x_2, y_2)\).

Step 2: Identify Coordinates

Identify the coordinates from the points given. For point \(P\), we have \(x_1 = -5\) and \(y_1 = 1\). For point \(Q\), we have \(x_2 = 3\) and \(y_2 = -2\).

Step 3: Substitute into the Formula

Substitute the identified coordinates into the slope formula: \( m = \frac{-2 - 1}{3 - (-5)} \).

Step 4: Calculate the Numerator

Calculate the difference in the y-coordinates (the numerator): \(-2 - 1 = -3\).

Step 5: Calculate the Denominator

Calculate the difference in the x-coordinates (the denominator): \(3 - (-5) = 3 + 5 = 8\).

Step 6: Compute the Slope

Divide the numerator by the denominator to get the slope: \( m = \frac{-3}{8} \).

Key Concepts

Slope Formula

The slope formula is a crucial tool in understanding how steep a line is on a graph. It is defined as the change in the y-values divided by the change in the x-values between two points. This is captured in the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) represent any two distinct points on a line. When using the slope formula, the result tells you how many units the line rises or falls vertically for each unit it moves horizontally. A positive slope indicates the line ascends, while a negative slope means it descends. If the slope equals zero, the line is horizontal; if it's undefined, the line is vertical. Using the slope formula correctly allows you to determine the nature and direction of a line at a glance.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves using algebraic equations to describe geometric principles. It connects algebra to geometry through the use of coordinates on a plane.In coordinate geometry, a point is defined by an ordered pair \((x, y)\). These coordinates represent its position on the two-dimensional plane:

• The first number, \(x\), is the point's horizontal position, or how far it is along the x-axis.
• The second number, \(y\), denotes the point's vertical position along the y-axis.

We use these coordinates to find distances, midpoints, and slopes - such as with our exercise involving points \(P(-5, 1)\) and \(Q(3, -2)\).Coordinate geometry allows us to calculate the slope by selecting any two points on a line. From these, we determine how the line behaves and looks graphically. This method is indispensable in geometry and calculus, where understanding the properties of lines and curves is fundamental.

Linear Equations

Linear equations describe lines on a graph and have the standard form \(y = mx + b\), where:\

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, or the point where the line crosses the y-axis.

This equation summarizes the entire line in a compact form, allowing you to understand its characteristics quickly.To comprehend the role of the slope \(m\), note that this value directly affects the angle of the line. It tells you how steep the line is.
If we know the slope, we can use it along with one point on the line to construct the full linear equation. This is essential for graphing the line or comparing it with other lines.In problem-solving, finding the slope or applying it within a linear equation is common. This helps solve numerous mathematical problems, including predicting unknown values, modeling real-world scenarios, and understanding trends.