Problem 66 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (-2,2),(-5,1) $$

Step-by-Step Solution

Answer

The slope of the line is \(\frac{1}{3}\).

Step 1: Identify the Coordinates

The first point is (-2, 2) and the second point is (-5, 1). Let's denote them as \((x_1, y_1) = (-2, 2)\) and \((x_2, y_2) = (-5, 1)\).

Step 2: Recall the Slope Formula

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

Step 3: Insert the Coordinates into the Formula

Substitute the coordinates into the formula: \[ m = \frac{1 - 2}{-5 + 2} \].

Step 4: Calculate the Differences

Calculate \(y_2 - y_1\) which is \(1 - 2 = -1\) and \(x_2 - x_1\) which is \(-5 + 2 = -3\).

Step 5: Calculate the Slope

Now, calculate the slope: \[ m = \frac{-1}{-3} = \frac{1}{3} \].The slope of the line is \(\frac{1}{3}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It allows us to describe geometric figures using algebra through the use of a coordinate plane, which consists of two perpendicular lines: the x-axis and the y-axis. This plane is how we precisely describe the position of points, lines, and shapes.
In coordinate geometry:

• Each point is identified by a pair of numbers, called coordinates. The first number is the x-coordinate and the second is the y-coordinate.
• Points are expressed as \( (x, y) \).
• Distance between two points and the slope of a line connecting them can be easily calculated using mathematical formulas.

The importance of coordinate geometry lies in its ability to connect algebra with geometry, enabling us to solve complex geometric problems using algebraic equations.

Linear Equations

A linear equation is an equation involving only first-degree terms. In the context of a line on a coordinate plane, it can be expressed in the form \( y = mx + b \) where \( m \) is the slope of the line, and \( b \) is the y-intercept. This equation describes a straight line that graphically represents the relationship between the x and y coordinates.

Linear equations are useful because:

• They make it easier to describe the path of a straight line.
• The slope, \( m \), indicates the line's steepness. A positive slope means the line ascends as it moves left to right, while a negative slope means it descends.
• The y-intercept, \( b \), tells us where the line crosses the y-axis.

Understanding linear equations is essential for studying and predicting behaviors in fields where relationships between variables matter, such as economics and physics.

Mathematical Formula

Mathematical formulas are equations designed to calculate or solve specific mathematical problems. In the context of finding the slope of a line, a key formula is the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula is derived from the ratio of the "rise" (the difference in y-values) over the "run" (the difference in x-values).
This mathematical formula has several significant advantages:

• It clearly defines the slope, which quantifies how steep a line is.
• Using this formula makes it easy to determine if two lines are parallel (identical slopes) or perpendicular (slopes that are negative reciprocals).
• It is a foundational tool in many areas of mathematics and engineering where analyzing linear relationships is essential.

Learning to apply mathematical formulas appropriately is crucial because they provide efficient and accurate ways to solve problems and understand the world through a mathematical lens.