Problem 87 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (-2,8),(1,-7) $$

Step-by-Step Solution

Answer

The slope of the line is -5.

Step 1: Identify the Points

We have two points given: Point 1: \((-2, 8)\)Point 2: \((1, -7)\)

Step 2: Recall the Slope Formula

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

Step 3: Substitute Points into the Slope Formula

Substitute Point 1 \((-2, 8)\) as \((x_1, y_1)\) and Point 2 \((1, -7)\) as \((x_2, y_2)\) into the slope formula:\[m = \frac{-7 - 8}{1 - (-2)}\]

Step 4: Simplify the Numerator

Calculate the change in the \(y\)-coordinates:Numerator: \(-7 - 8 = -15\)

Step 5: Simplify the Denominator

Calculate the change in the \(x\)-coordinates:Denominator: \(1 - (-2) = 1 + 2 = 3\)

Step 6: Calculate the Slope

Now, substitute the simplified numerator and denominator back into the slope formula:\[m = \frac{-15}{3}\]Then simplify:\[m = -5\]

Key Concepts

Understanding the Slope Formula

The slope formula is a fundamental concept in mathematics that helps us determine how steep a line is between two points on a graph. If you imagine a hill, the slope tells us how steep that hill is.

The formula itself is straightforward: it's represented as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where:

• \( m \) is the slope of the line.
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points on the line.
• \((y_2 - y_1)\) represents the change in the vertical direction, often referred to as the "rise."
• \((x_2 - x_1)\) represents the change in the horizontal direction, often referred to as the "run."

The slope essentially measures the "rise over run."
It's a way of expressing how much and in which direction you need to move to get from one point to the other along the line. A positive slope means the line goes upwards as it moves from left to right, while a negative slope means it goes downwards.
Knowing how to calculate and interpret the slope is crucial for understanding the behavior of linear functions and analyzing real-world situations.

Introduction to Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry that uses a coordinate system to explore geometric problems. By using the x and y-axes to plot points, we can leverage algebra to find distances, slopes, and other geometric properties.

This system helps us better visualize and solve problems involving shapes and lines. For example, when given two points, understanding their position on a coordinate plane helps us calculate the line's slope, distance, or even equation.

• The x-axis runs horizontally, and is where we plot horizontal movements.
• The y-axis runs vertically, for vertical movements see.
• Each point on the plane is defined by an ordered pair \((x, y)\), representing its horizontal and vertical distances from the origin \((0,0)\).

Coordinate geometry serves as a powerful tool to bridge the gap between algebra and geometry, making it easier to solve complex problems by breaking them down into simpler algebraic equations.

Decoding Linear Equations

Linear equations are algebraic expressions that depict a straight line when graphed on a coordinate plane. They take the form \(y = mx + b\), where:

• \(y\) is the dependent variable or the output of the equation.
• \(x\) is the independent variable or input.
• \(m\) denotes the slope of the line, revealing its steepness and direction.
• \(b\) is the y-intercept, indicating the point where the line crosses the y-axis.

Linear equations are called "linear" because their graph is a line. Understanding them is important for modeling and predicting behaviors in numerous practical settings, from economics to physics.

The relationship they describe between two variables is direct and proportional, making these equations a cornerstone of linear algebra. For example, in real-life scenarios, linear equations can model things like speed (distance over time) or cost (price per item).
By mastering linear equations, you gain the ability to simplify and resolve complex data patterns into understandable lines and straight-forward solutions.