Question
f \( y \) varies directly with \( x \) and \( y=6 \) when \( x=3 \), find \( y \) when \( x=2 \). Write and solve a direct variation equation to find the answer. \( y=\square \) Submit
Ask by Henry Pearson. in the United States
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( y = 4 \)
Solution
To solve the direct variation problem, follow these steps:
### 1. **Understand Direct Variation**
When \( y \) varies directly with \( x \), it means that \( y \) is equal to a constant multiplied by \( x \). This relationship can be expressed as:
\[
y = kx
\]
Where:
- \( k \) is the constant of variation.
### 2. **Find the Constant of Variation (\( k \))**
You're given that \( y = 6 \) when \( x = 3 \). Plug these values into the equation to find \( k \):
\[
6 = k \times 3
\]
Solving for \( k \):
\[
k = \frac{6}{3} = 2
\]
### 3. **Write the Direct Variation Equation**
Now that you have the value of \( k \), substitute it back into the direct variation equation:
\[
y = 2x
\]
\[
\boxed{y = 2x}
\]
### 4. **Find \( y \) When \( x = 2 \)**
Use the direct variation equation to find \( y \) when \( x = 2 \):
\[
y = 2 \times 2 = 4
\]
\[
\boxed{y = 4}
\]
### **Summary**
- **Direct Variation Equation:** \( y = 2x \)
- **When \( x = 2 \):** \( y = 4 \)
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Simplify this solution The Deep Dive
To find the direct variation equation, start by expressing the relationship between \( y \) and \( x \) as \( y = kx \), where \( k \) is the constant of variation. Given \( y = 6 \) when \( x = 3 \), you can calculate \( k \) by rearranging the equation: \[ k = \frac{y}{x} = \frac{6}{3} = 2. \] Now, your direct variation equation is \( y = 2x \). To find \( y \) when \( x = 2 \), substitute \( 2 \) for \( x \): \[ y = 2(2) = 4. \] So, \( y = 4 \). \( y = 4 \)
