Problem 52 - Algebra 1

Question

Find the rate of change between the two points. Give the units of measure for the rate. \((2,2)\) and \((9,23) ; x\) in minutes, \(y\) in inches

Step-by-Step Solution

Answer

The rate of change between the two points is 3 inches per minute.

Step 1: Identify the Coordinated Points

We have already been given two coordinates. Identify the two points as (x1, y1) = (2, 2) and (x2, y2) = (9, 23).

Step 2: Apply Slope Formula

Slope is defined as the change in vertical values divided by the change in horizontal values, which is calculated as: slope (rate of change) = (y2 - y1) / (x2 - x1).

Step 3: Calculate the Slope/Rate of Change

Substitute the coordinated values into the slope formula: (23 - 2) / (9 - 2) = 21 / 7 = 3.

Step 4: Define the Units of Measurement

Since the x-value is in minutes and the y-value is in inches, the rate of change (slope) would be measured in inches per minute.

Key Concepts

Slope Calculation

To understand how to calculate the slope, known also as the rate of change, let's break it down step-by-step. Imagine you have two coordinate points on a graph. These points help in determining how steep a line is between them. The formula for slope is very simple:\[\text{Slope} = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are your two points. The slope measures the steepness of a line, showing how much \( y \) (vertical change) changes per unit of \( x \) (horizontal change). This aspect of stretch or compression visually can be quite useful in understanding how different quantities relate to each other. In the example given:

• Point 1: \( (2, 2) \)
• Point 2: \( (9, 23) \)

Plug these into the formula: \[\frac{23 - 2}{9 - 2} = \frac{21}{7} = 3\]This calculation shows that for each minute increase in time, the change is 3 inches. This value of 3 denotes the slope.

Units of Measurement

Units of measurement are crucial when discussing rates of change. They offer clarity about the quantities being compared. In problems involving slope, understanding units helps interpret the physical meaning of the slope value.In the exercise, the given points come with units:

• \( x \) - representing time, measured in **minutes**
• \( y \) - representing distance, measured in **inches**

Calculating the slope involves not only numbers; it's essential to attach the units. Thus, the slope of 3 becomes meaningful when we say it measures **3 inches per minute**. This tells us how fast the distance in inches changes with time, enhancing comprehension by attaching a unit-driven context.

Coordinate Points

Coordinate points are pairs of values that show a specific location on a graph. These consist of an \( x \)-value and a \( y \)-value. It's like a map for your data, telling you exactly where each point is situated.In reference to our problem, we've got two pivotal points that create our line:

• \( (2, 2) \)
• \( (9, 23) \)

• The \( x \)-value \((2, 9)\) signifies points on the horizontal axis, each representing minutes.
• The \( y \)-value \((2, 23)\) is on the vertical axis, each representing inches.

Coordinate points act as the foundation for calculating any change or rate the line might experience. By understanding how each point contributes to defining a line's details, you empower your analytical skills to make accurate interpretations about data. This understanding bridges the gap between mere points and the dynamic line they compose on your graph.