Problem 89 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (4,21),(9,12) $$

Step-by-Step Solution

Answer

The slope is \(-\frac{9}{5}\).

Step 1: Identify Points

First, identify the given points. The points provided are \((x_1, y_1) = (4, 21)\) and \((x_2, y_2) = (9, 12)\).

Step 2: Use the Slope Formula

Use the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula calculates the slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\).

Step 3: Substitute Values

Substitute the values from the points into the slope formula: \( m = \frac{12 - 21}{9 - 4} \).

Step 4: Calculate Differences

Calculate the difference in the y-values: \( 12 - 21 = -9 \), and the difference in the x-values: \( 9 - 4 = 5 \).

Step 5: Simplify the Slope

Simplify the expression \( m = \frac{-9}{5} \). The slope of the line is \(-\frac{9}{5}\).

Key Concepts

Points Identification

To find the slope of a line between two points, the first step involves identifying the coordinates given in the problem. In this exercise, the points provided are

• \((4, 21)\) which represents \((x_1, y_1)\)
• \((9, 12)\) which represents \((x_2, y_2)\)

This identification is crucial because it sets up the use of the slope formula later on.Next, these points represent specific locations on a graph, where each point has an x-coordinate and a y-coordinate. Identifying these correctly ensures precise calculations in the following steps.

Slope Formula

The slope of a line defines its steepness and direction. To determine the slope, we use a specific formula. This formula is crucial because it provides a consistent way to calculate slope for any pair of points. The slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( m \) represents the slope of the line. The formula essentially measures the "rise" (change in y-values) over the "run" (change in x-values). This mathematical tool is essential in geometry and helps in understanding the relationship between two points on a line.

Substitution in Formula

With the points identified and the slope formula ready, the next step is to substitute the actual coordinates into the formula. Substitution is straightforward if the points are labeled correctly.Plug the coordinates from the points into our formula \[ m = \frac{12 - 21}{9 - 4} \] This substitution is a simple mathematical operation but it requires careful attention to detail. A minor mistake in handling the coordinates could lead to an incorrect slope, so it's important to be precise and check the calculations.

Calculating Differences

After placing the values into the slo

Calculating Differences

After placing the values into the slope formula, the next step involves calculating the differences in y-values and x-values of the given points. The subtraction operations should be handled step-by-step:

• The difference between \( y \) values: \( 12 - 21 = -9 \)
• The difference between \( x \) values: \( 9 - 4 = 5 \)

These differences represent how much the line rises or falls as it moves horizontally in the graph. Ensuring correct calculation here is vital before simplifying the expression.

Simplifying Expressions

Once the differences for the y-values and x-values are calculated, the final step is to simplify this ratio to find the slope. In this exercise, you end up with: \[ m = \frac{-9}{5} \] This step is essential as it provides the final slope in its simplest form. The simplified slope, \(-\frac{9}{5}\), indicates the line descends, or falls, as you move from left to right across the graph. By simplifying, we ensure the slope is readily interpretable, displaying the relationship between the two points with clarity.