Problem 88 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (3,8),(7,22) $$

Step-by-Step Solution

Answer

The slope of the line is \( \frac{7}{2} \).

Step 1: Understand the Slope Formula

To find the slope of a line passing through two points, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Step 2: Identify the Point Coordinates

Identify the coordinates of the points given. Here, the first point is \((x_1, y_1) = (3, 8)\) and the second point is \((x_2, y_2) = (7, 22)\).

Step 3: Calculate the Difference in Y-coordinates

Subtract the y-coordinate of the first point from the y-coordinate of the second point: \( y_2 - y_1 = 22 - 8 = 14 \).

Step 4: Calculate the Difference in X-coordinates

Subtract the x-coordinate of the first point from the x-coordinate of the second point: \( x_2 - x_1 = 7 - 3 = 4 \).

Step 5: Apply the Slope Formula

Substitute the differences into the slope formula: \( m = \frac{14}{4} \).

Step 6: Simplify the Slope

Simplify the fraction to get the slope: \( m = \frac{14}{4} = \frac{7}{2} \).

Key Concepts

Slope Formula

The slope of a line is a measure of its steepness and direction. To find the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we use a simple formula:

• The slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula helps determine how much the line 'rises' vertically as it 'runs' horizontally between two points. This is why it is often referred to as 'rise over run'.
When using this formula, make sure to:

• Consistently use two points identified as \( (x_1, y_1) \) and \( (x_2, y_2) \).
• Subtract the first point's coordinates from the second point's coordinates in the correct order.

In our example, the slope \( m = \frac{14}{4} = \frac{7}{2} \), meaning for every 2 units moved sideways, the line goes up 7 units.

Coordinate Geometry

Coordinate geometry, or analytical geometry, involves studying geometric figures using a coordinate system. Points in this system are identified using ordered pairs, providing a way to examine shapes numerically.
In problems like the one we're exploring, each point \( (x, y) \) contains:

• \( x \) being the horizontal position on the x-axis.
• \( y \) being the vertical position on the y-axis.

When calculating the slope of a line, we observe changes in these \( x \) and \( y \) values. This change helps describe the relationship between the two points geometrically and numerically.
Understanding coordinate geometry allows us to visualize the linear relationship in space, making abstract algebraic concepts more tangible and easier to analyze.

Algebra Concepts

Algebra provides the foundational language for expressing relationships between numbers and variables. In the case of calculating the slope:

• We use variables like \( x_1, x_2, y_1, \text{and} y_2 \) to represent specific coordinates from the problem context.

Algebra involves manipulating these variables according to set rules. For finding the slope:- **Difference Calculation:** Calculate \( y_2 - y_1 \) and \( x_2 - x_1 \). This step is key in using algebra to determine the change between these variable values effectively.- **Expression Simplification:** After substituting into the slope formula, simplify \( \frac{y_2 - y_1}{x_2 - x_1} \). In our example, simplifying \( \frac{14}{4} \) to \( \frac{7}{2} \, \) embodies a core algebraic process.Algebra allows these relationships and calculations to be consistent and reliable, providing a means to analyze problems systematically.