Problem 85 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (-3,-2),(-1,-4) $$

Step-by-Step Solution

Answer

The slope of the line is -1.

Step 1: Identify the Points

The points given in the problem are \((-3, -2)\) and \((-1, -4)\). We will label them as \((x_1, y_1) = (-3, -2)\) and \((x_2, y_2) = (-1, -4)\).

Step 2: Know the Slope Formula

The formula to calculate the slope \(m\) of a line passing through two 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 into the Slope Formula

Substitute the known values into the formula: \[ m = \frac{-4 - (-2)}{-1 - (-3)} \].

Step 4: Simplify the Differences

Perform the operations inside the formula: In the numerator, \(-4 - (-2) = -4 + 2 = -2\), and in the denominator, \(-1 - (-3) = -1 + 3 = 2\).

Step 5: Calculate the Slope

Plug the simplified values back into the slope formula: \[ m = \frac{-2}{2} = -1 \].

Key Concepts

Point-Slope Formula

The point-slope formula is a handy tool in coordinate geometry that relates a point on a line to its slope. It allows us to write the equation of a line when a point \(x_1, y_1\) on the line and the slope \(m\) are known. The formula itself is expressed as: \[ y - y_1 = m(x - x_1) \].

• Point: This is any specific point on the line, defined by its coordinates (e.g., \((-3, -2)\)).
• Slope (m): This is the measure of the steepness or incline of a line, calculated by the change in y-coordinates divided by the change in x-coordinates between two distinct points.

Using this formula, once you know the slope and at least one point on the line, you can easily find the equation of the line. This is particularly useful in problems that involve predicting values or determining the line's behavior over different points.

Linear Equations

Linear equations are algebraic expressions of the form \(ax + by = c\), where the graph is a straight line. They are fundamental in understanding the relationship between two variables that change at a constant rate. For graphing, linear equations can be expressed in different forms, including:

• Slope-Intercept Form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• Point-Slope Form: \(y - y_1 = m(x - x_1)\), connecting the slope and a specific point on the line.

Linear equations are vital because they describe real-world situations where one quantity depends on another. For example, they can be used to model financial predictions, track speed over time, or determine trends in data.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebra to investigate geometric properties. It allows us to represent geometric shapes in a numerical way through the coordinate plane.

• Coordinate Plane: This is a two-dimensional surface where points are located using an ordered pair of numbers \(x, y\).
• Lines: In coordinate geometry, lines are the simplest figures that can be graphed. Understanding their equations helps in visualizing and solving geometric problems.

Using coordinates and the relationships between them, coordinate geometry supports problem-solving in various fields including physics, engineering, and computer science. Students will find that grasping the basics of coordinate geometry gives them powerful tools for analyzing written data and graphs effectively.