Problem 20 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((4,-5)\) and \((-2,3)\)

Step-by-Step Solution

Answer

The slope is approximately \ -1.33 \.

Step 1 - Identify the coordinates

Identify the given points as \((x_1, y_1) = (4, -5)\) and \((x_2, y_2) = (-2, 3)\).

Step 2 - Use the slope formula

The formula for the slope (m) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is \[m = \frac{y_2 - y_1}{x_2 - x_1}\].

Step 3 - Substitute the values

Substitute \(x_1, y_1, x_2, y_2\) into the slope formula: \[m = \frac{3 - (-5)}{-2 - 4}\].

Step 4 - Simplify the expression

Simplify the numerator and the denominator: \[m = \frac{3 + 5}{-2 - 4} = \frac{8}{-6} = -\frac{4}{3}\].

Step 5 - Round the result

Convert \(-\frac{4}{3}\) to a decimal and round to the nearest hundredth: \[ m \approx -1.33 \].

Key Concepts

slope formula

The slope formula is a key tool in coordinate geometry. It's used to calculate the steepness or incline of a line. Given two points on a line, we can determine the slope using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1} \] The variables \(x_1, y_1\) and \(x_2, y_2\) represent the coordinates of the two points. Essentially, this formula measures how much the \(y\) values change per unit change in \(x\) values, capturing the line's steepness. For instance, the slope between the points \((4, -5)\) and \((-2, 3)\) is calculated by substituting these coordinates into the slope formula: \[ m = \frac{3 - (-5)}{-2 - 4} \] Simplify the expression to find the slope: \[ m = \frac{8}{-6} = -\frac{4}{3}\] Which can be approximated in decimal form: \[ m \approx -1.33\] This means for every unit increase in \(x\), the \(y\) value decreases by approximately 1.33 units.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics bridges algebra and geometry. Points on a plane are represented by ordered pairs of numbers, \((x, y)\), known as coordinates. These pairs tell us about the horizontal ( \(x\) ) and vertical ( \(y\)) positions on the coordinate plane. When we discuss lines in this context, we often use the concept of slope to describe their direction and steepness. The basic concepts of coordinate geometry include:

• Coordinates: Representing points using pairs \((x, y)\).
• Distance Formula: Calculating the distance between two points.
• Midpoint Formula: Finding the middle point between two given points.
• Slope Formula: Determining the inclination of a line.

algebra

Algebra involves the study of mathematical symbols and the rules for manipulating these symbols. It is a unifying thread of almost all of mathematics. In the context of the slope formula, algebraic manipulation allows us to simplify the expressions and solve for the desired variable. For example, consider the calculation of slope: Start with the slope formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Substitute the given points \((4, -5)\) and \((-2, 3)\): \[ m = \frac{3 - (-5)}{-2 - 4}\] Simplify the numerator and denominator separately: \[ m = \frac{3 + 5}{-2 - 4} = \frac{8}{-6} = -\frac{4}{3}\] Here, algebra helps us combine and reduce fractions for a neat solution. Lastly, converting \( -\frac{4}{3} \) to a decimal gives \( -1.33 \), demonstrating the application of algebra in real-world problems.