Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Find the slope (if defined) of the line that passes through the given points. $$(-2,1) \text { and }(3,6)$$

Step-by-Step Solution

Answer

The slope of the line is 1.

Step 1: Understand the Slope Formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula calculates the change in y over the change in x, representing how steep the line is.

Step 2: Identify Coordinates

Identify the coordinates from the problem. The first point is \((-2, 1)\) and the second point is \((3, 6)\). Let \((x_1, y_1) = (-2, 1)\) and \((x_2, y_2) = (3, 6)\).

Step 3: Substitute Coordinates into Slope Formula

Substitute the values into the slope formula: \[ m = \frac{6 - 1}{3 - (-2)} \].

Step 4: Simplify the Fraction

Calculate the changes in y and x: the change in y is \(6 - 1 = 5\) and the change in x is \(3 - (-2) = 3 + 2 = 5\). So the slope \( m \) can be simplified to \[ m = \frac{5}{5} \].

Step 5: Simplify the Expression

Simplify the fraction \(\frac{5}{5}\) which results in \(1\). Therefore, the slope of the line is \(m = 1\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra with geometry through the use of a coordinate plane. In this system, we describe points with ordered pairs \((x, y)\) that indicate horizontal (x) and vertical (y) positions. Each point's coordinate acts as a bridge between numerical calculations and geometric understanding.

A line in coordinate geometry is defined by such points and is a straight path extending both ways forever. When two distinct points are given, we can create a line by understanding their x and y coordinates.

• A coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
• Points on this plane are written as \((x, y)\), where x is the position along the horizontal axis and y is the height above or below the origin.
• These coordinates enable us to perform calculations such as finding the slope of a line that intersects them.

When working with coordinate geometry, it's important to interpret these points effectively to draw diagrams or solve problems like calculating slope.

Linear Equations

Linear equations are mathematical expressions that model a line within a coordinate plane. They take the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

The slope of a line is crucial because it describes the line's steepness and direction. It is calculated as the ratio of vertical change to horizontal change between two points, helping us understand how the y-values change as the x-values increase or decrease.

• A constant slope means the rate of change is steady, leading to a straight line on the graph.
• Positive slopes indicate lines rising from left to right, while negative slopes suggest a descent.

In the given problem, the linearity implies that every equal increase in x results in the same increase in y. Understanding the linear relationship prepares us to solve for unknown variables and graph these relationships accurately.

Mathematical Formula

Mathematical formulas are essential tools that provide a consistent and reliable method to calculate desired values. In the context of finding a line's slope, the slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This expression helps in determining how one quantity changes in relation to another.

When using this formula, follow these steps:

• Identify the coordinates \((x_1, y_1)\) and \((x_2, y_2)\) from the points given.
• Calculate the difference in y-values \((y_2 - y_1)\) to find the rise.
• Compute the difference in x-values \((x_2 - x_1)\) to determine the run.
• Divide the rise by the run to get the slope \(m\).

In our example, by substituting the points \((-2, 1)\) and \((3, 6)\) into the formula, we see that both the rise and run equal \(5\), giving a slope \(m = 1\). This showcases how mathematical formulas bridge numerical data and geometric layouts, ensuring precise calculations.