the relationship between x and y is given by the equation 3x - 4y = 12 and the range of y is 9 , then the range of x is?

To find the range of \(x\) given the equation \(3x - 4y = 12\) and the range of \(y\), we need to understand the relationships and constraints provided.

### Step-by-Step Solution

1. **Understand the given equation**:

   The equation provided is:
   \[
   3x - 4y = 12
   \]

2. **Rearrange the equation to express \(x\) in terms of \(y\)**:

   Solving for \(x\), we get:
   \[
   3x = 4y + 12
   \]
   \[
   x = \frac{4y + 12}{3}
   \]

3. **Identify the range of \(y\)**:

   The range of \(y\) is given as 9. This means that \(y\) can take on values within a specific interval of length 9. However, without explicit bounds for \(y\), we consider a general interval \([a, b]\) such that \(b - a = 9\).

4. **Determine the range of \(x\) based on \(y\)'s range**:

   Let's say \(y\) ranges from \(a\) to \(a + 9\). Then:

   - The minimum value of \(y\) is \(a\).
   - The maximum value of \(y\) is \(a + 9\).

   Substituting these into the equation for \(x\):

   \[
   x_{\text{min}} = \frac{4a + 12}{3}
   \]
   \[
   x_{\text{max}} = \frac{4(a + 9) + 12}{3} = \frac{4a + 36 + 12}{3} = \frac{4a + 48}{3}
   \]

5. **Compute the difference \(x_{\text{max}} - x_{\text{min}}\)**:

   \[
   x_{\text{max}} - x_{\text{min}} = \frac{4a + 48}{3} - \frac{4a + 12}{3}
   \]
   \[
   = \frac{(4a + 48) - (4a + 12)}{3}
   \]
   \[
   = \frac{48 - 12}{3}
   \]
   \[
   = \frac{36}{3} = 12
   \]

Therefore, **the range of \(x\) is 12**.

### Conclusion

Given the equation \(3x - 4y = 12\) and that the range of \(y\) is 9, the range of \(x\) is 12.

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Would you like more details, or do you have any questions about this solution?

Here are some related questions to consider:

1. How would the range of \(x\) change if the range of \(y\) were different?
2. What happens to the range of \(x\) if the equation is modified to \(5x - 4y = 12\)?
3. Can the range of \(x\) be negative if \(y\) has a positive range?
4. How does changing the coefficients in the equation \(3x - 4y = 12\) affect the ranges of \(x\) and \(y\)?
5. What would be the effect on the range of \(x\) if \(y\) had an upper and lower bound other than a fixed range?

**Tip**: Always double-check your algebraic manipulations when solving for one variable in terms of another, especially when dealing with ranges and inequalities.

Explore how to determine the range of x given a linear equation 3x - 4y = 12 and a specified range for y. Detailed step-by-step solution and calculations included. Suitable for Grades 7-9.

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