Problem 25 - Algebra 1: Concepts and Skills

Question

FINDING SLOPE Find the slope of the line that passes through the points. \((3,8)\) and \((7,7)\)

Step-by-Step Solution

Answer

The slope of the line that passes through the points (3,8) and (7,7) is -1/4.

Step 1: Identify the Given Points

Identify and label the given points. The first point is, (3,8), so \(x1 = 3\) and \(y1 = 8\). The second point is, (7,7), so \(x2 = 7\) and \(y2 = 7\).

Step 2: Apply the Slope Formula

Insert the values into the slope formula. So, \(m = \frac{{7-8}}{{7-3}} = -\frac{1}{4}\).

Step 3: Solution

The slope of the line that passes through the points (3,8) and (7,7) is -1/4.

Key Concepts

Slope Formula

Understanding the slope of a line is fundamental in algebra, especially when dealing with linear equations. The slope is a measure of how steep a line is on a graph. To find the slope, also known as the rate of change, we use the slope formula:

\[\begin{equation}m = \frac{y2 - y1}{x2 - x1}\end{equation}\]
Here, the letter 'm' represents the slope, and (x1, y1), (x2, y2)are coordinate points on the line with x representing the horizontal axis and y representing the vertical axis. The formula takes the difference in the y-values divided by the difference in the x-values of these two points. The resulting fraction will tell us how much the line rises (positive slope) or falls (negative slope) for every unit move along the x-axis. In the given exercise, we find the slope is negative, which indicates the line is falling as it moves from left to right on the graph.

Coordinate Points

Coordinate points are the building blocks for graphing lines and shapes in a two-dimensional plane. Each point is defined by a pair of numbers, (x, y),where 'x' is the position relative to the horizontal (x) axis and 'y' is the position relative to the vertical (y) axis. These numbers can represent a variety of things depending on the context, such as distance, time, or other variables.

Identifying and plotting these points accurately is crucial for analyzing and interpreting graphical data correctly. To solve for the slope as in our exercise, we referred to the coordinate points (3,8) and (7,7). We designated these points as (x1, y1)and (x2, y2)respectively, which formed the basis for our calculations when applying the slope formula.

Linear Equations

Linear equations form the foundation of a range of mathematical concepts and are frequently encountered in many fields. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and the single power of a variable. These can be graphically represented as straight lines in a coordinate plane.

The General Form

A linear equation generally comes in the form: \[\begin{equation}y = mx + b\end{equation}\]
where 'm' stands for the slope, and 'b' represents the y-intercept, which is where the line crosses the y-axis. The slope, as previously explained, tells us about the inclination or steepness of the line.

Linear equations are used to model real-world phenomena, solve systems of equations, and are the first step to understanding more complex mathematical concepts. By knowing how to find a line's slope, such as we did with the points (3,8) and (7,7), we can not only sketch the line but also comprehend its behavior and relationship between the variables being examined.