It looks like you’re looking for help with proportional relationships and their equations. Proportional relationships are those in which two quantities maintain a constant ratio to each other. Here’s a quick overview of how to identify and write equations for proportional relationships:
Key Concepts:
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Constant of Proportionality: The ratio between the two quantities in a proportional relationship is called the constant of proportionality (k). It can be expressed as:
\[ k = \frac{y}{x} \]
where \(y\) and \(x\) are the quantities involved.
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Equation of Proportional Relationships: Proportional relationships can be written as:
\[ y = kx \]
where \(y\) is the dependent variable, \(x\) is the independent variable, and \(k\) is the constant of proportionality.
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Graph of Proportional Relationships: The graph of a proportional relationship is a straight line that passes through the origin (0,0).
Steps to Identify and Write Equations:
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Determine if the Relationship is Proportional: Check if the ratio between the two quantities is constant.
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Find the Constant of Proportionality: Calculate \(k\) using any pair of (x, y) values.
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Write the Equation: Substitute \(k\) into the equation \(y = kx\).
Example Problem:
Problem: A recipe requires 3 cups of flour for every 4 cups of sugar. Is this a proportional relationship? If so, write the equation.
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Determine if it’s Proportional: The ratio of flour to sugar is consistent.
\( \frac{3 \text{ cups of flour}}{4 \text{ cups of sugar}} = \frac{3}{4}\)
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Find the Constant of Proportionality: Here, we can express the relationship as \(y = kx\) where \(y\) could be the amount of flour and \(x\) is sugar. \[ k = \frac{3 \text{ cups of flour}}{4 \text{ cups of sugar}} = \frac{3}{4} \]
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Write the Equation: Substituting in the values, you get: \[ y = \frac{3}{4}x \]
If you have specific questions or problems you'd like to go over, feel free to share them!