Solving linear functions with modeling involves representing real-world scenarios using linear equations, allowing you to analyze and predict outcomes. Here's a step-by-step guide:
Steps to Solve Linear Functions with Modeling
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Identify Input and Output Values: Determine the independent variable (input, often denoted as 'x') and the dependent variable (output, often denoted as 'y'). The independent variable is the factor that influences the dependent variable. For example, if you're modeling the cost of renting a car, the number of miles driven might be the input (x), and the total cost would be the output (y).
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Convert Data to Coordinate Pairs: Translate the given information into coordinate pairs (x, y). Each pair represents a specific instance of the relationship between the input and output. Using the car rental example, if renting the car for 50 miles costs \$75, one coordinate pair would be (50, 75). You need at least two coordinate pairs to define a linear function.
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Find the Slope: Calculate the slope (m) of the line using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) and (x₂, y₂) are two coordinate pairs. The slope represents the rate of change of the output with respect to the input. In the car rental example, the slope might represent the cost per mile driven.
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Write the Linear Model: Use the slope-intercept form of a linear equation:
y = mx + b
where 'm' is the slope, and 'b' is the y-intercept (the value of y when x is 0). To find 'b', substitute the slope (m) and one of the coordinate pairs (x, y) into the equation and solve for 'b'. The y-intercept represents the fixed cost, regardless of the input variable.
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Use the Model for Predictions: Once you have the linear equation, you can use it to make predictions. Substitute a given x-value into the equation and solve for y. This will give you the predicted output for that specific input. For instance, if you want to know the cost of driving 100 miles, substitute x = 100 into the equation.
Example
Let's say you're modeling the height of a plant over time.
- Day 1 (x = 1): Height = 2 cm (y = 2)
- Day 3 (x = 3): Height = 6 cm (y = 6)
- Input and Output: Input = Day (x), Output = Height (y)
- Coordinate Pairs: (1, 2) and (3, 6)
- Slope: m = (6 - 2) / (3 - 1) = 4 / 2 = 2 cm/day
- Linear Model: y = 2x + b. Using point (1,2): 2 = 2(1) + b, so b = 0. Therefore, y = 2x
- Prediction: To predict the height on Day 5, substitute x = 5: y = 2(5) = 10 cm
Table Summarizing the Process
| Step | Description | Example (Plant Growth) |
|---|---|---|
| Identify Input/Output | Determine independent (x) and dependent (y) variables. | Day (x), Height (y) |
| Convert to Coordinate Pairs | Represent data as (x, y) pairs. | (1, 2), (3, 6) |
| Find the Slope | Calculate the rate of change (m). m = (y₂ - y₁) / (x₂ - x₁) | m = 2 cm/day |
| Write Linear Model | Use slope-intercept form (y = mx + b) to define the relationship. | y = 2x |
| Make Predictions | Substitute x-value into the equation to find corresponding y-value. | Day 5: y = 10 cm |
By following these steps, you can effectively model and solve real-world problems using linear functions.
