Problem 11 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((4,7)\) and \((-3,7)\)

Step-by-Step Solution

Answer

The slope is 0.

Step 1: Understand the Slope Formula

The formula to calculate the slope of a line passing through two points \((x_1, y_1)\text{ and }(x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 2: Identify the Coordinates

Identify the given points: \((x_1, y_1) = (4, 7)\) and \((x_2, y_2) = (-3, 7)\).

Step 3: Substitute the Values into the Formula

Substitute the values into the slope formula: \ \[ m = \frac{7 - 7}{-3 - 4} = \frac{0}{-7} \].

Step 4: Simplify

Simplify the expression: \ \[ m = 0 \].

Key Concepts

coordinate geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to study the positions and properties of geometric figures. It uses a coordinate system, usually composed of a set of two or three axes to pinpoint locations.

The most common system is the rectangular coordinate system, where any point can be described by a pair of coordinates \(x, y\). In our problem, we use this system to determine the position of two points \(4,7\) and \(-3,7\).

Calculating the slope between these points involves examining their coordinates and determining how the line connecting them changes as it moves from one point to the other. This is fundamental in understanding various aspects of a geometric figure in terms of location and size.

linear equations

Linear equations are equations that form a straight line when graphed. The general form of a linear equation in two variables is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

The slope \(m\) describes how steep the line is and can be calculated by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1} \). In our exercise, the points given are \( (4, 7)\) and \(-3, 7\).

By substituting the coordinates into the slope formula, we found that \(m = 0\). This indicates that the points lie on a horizontal line, reflecting no vertical change as one moves along the x-axis. Understanding linear equations and their slopes helps in predicting and describing linear relationships between variables.

simplifying expressions

Simplifying expressions involves reducing complex mathematical expressions into their simplest form. This step is essential in making our calculations and results more comprehensible.

In the given exercise, after substituting the coordinates into the slope formula, we have the expression: \( m = \frac{7 - 7}{-3 - 4} \).

Performing the arithmetic inside the numerator and the denominator gives us \( m = \frac{0}{-7} \). Simplifying this expression further yields \( m = 0 \).

Simplifying expressions allows us to arrive at a clear and understandable result, facilitating a deeper understanding of the mathematical concepts in play. In this exercise, determining that the slope \(m = 0\) quickly reveals the nature of the line connecting the points: it is perfectly horizontal.