Question

Equations of Proportional Relationships Quick Check

Answer 1

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:

1. **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.

2. **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.

3. **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:

1. **Determine if the Relationship is Proportional**: Check if the ratio between the two quantities is constant.

2. **Find the Constant of Proportionality**: Calculate \(k\) using any pair of (x, y) values.

3. **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.

1. **Determine if it’s Proportional**: The ratio of flour to sugar is consistent.

\( \frac{3 \text{ cups of flour}}{4 \text{ cups of sugar}} = \frac{3}{4}\)

2. **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.
\[
k = \frac{3 \text{ cups of flour}}{4 \text{ cups of sugar}} = \frac{3}{4}
\]

3. **Write the Equation**:
Substituting in the values, you get:
\[
y = \frac{3}{4}x
\]

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