Problem 24 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \((-3,5),(1,-6)\)

Step-by-Step Solution

Answer

The slope is \( -\frac{11}{4} \).

Step 1: Identify the coordinates

The given points are \((-3,5)\) and \((1,-6)\). Identify \(x_1, y_1\) for the first point and \(x_2, y_2\) for the second point. Here, \(x_1 = -3, y_1 = 5, x_2 = 1, y_2 = -6\).

Step 2: Use the slope formula

The formula to calculate the slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates into the formula gives: \[ m = \frac{-6 - 5}{1 - (-3)} \]

Step 3: Perform the calculations

First, compute the difference in the y-coordinates: \[ -6 - 5 = -11 \] Next, compute the difference in the x-coordinates: \[ 1 - (-3) = 1 + 3 = 4 \] Then, divide the difference in y-coordinates by the difference in x-coordinates: \[ m = \frac{-11}{4} \]

Step 4: Simplify the result

The simplified form of the slope is: \[ m = -\frac{11}{4} \]

Key Concepts

coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where points are defined and their relationships studied using a coordinate system. In this system, each point on the plane is described by a pair of numbers known as coordinates. The horizontal line is the x-axis, while the vertical line is the y-axis.

Every point in a coordinate plane has an x-coordinate (horizontal position) and a y-coordinate (vertical position). For instance, the point \((-3,5)\) has an x-coordinate of -3 and a y-coordinate of 5. The coordinates are usually written as \( (x, y) \). Understanding coordinate geometry is an essential foundation for solving problems involving the slope or equations of lines.

slope formula

The slope of a line indicates how steep the line is and the direction it goes. Mathematically, the slope is represented as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope formula is used to calculate this value and is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.

Let's break it down using our exercise:

• First, identify the coordinates: \(x_1 = -3\), \(y_1 = 5\), \(x_2 = 1\), \(y_2 = -6\).
• Next, substitute these values into the formula: \m = \frac{-6 - 5}{1 - (-3)} \.
• Finally, simplify the math to find the slope: \(m = \frac{-11}{4} = -\frac{11}{4}\).

In essence, the slope tells us that for every 4 units we move horizontally, the line moves down by 11 units, showing a negative slope.

linear equations

Linear equations represent lines on the coordinate plane. Each linear equation can be written in the standard form:

\[ y = mx + c \]

Where

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

To form the equation of a line between two points, first find the slope using the slope formula. With our example, the slope is \( -\frac{11}{4} \). If we need the equation, we can use one of the points \((x_1, y_1)\) along with the slope in the point-slope form of the linear equation:

\[ y - y_1 = m(x - x_1) \]

Substituting \((-3, 5)\) and using \( m = -\frac{11}{4} \), we write:

\[ y - 5 = -\frac{11}{4}(x + 3) \]

Simplify to get the standard form of the line equation. This approach helps in understanding the relationship of the slope, y-intercept, and the overall behavior of the line on a coordinate plane.