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### 6. \( 4x + 3y = 0 \)
- To determine if \( y \) varies directly with \( x \), we need to check if the equation can be written in the form \( y = kx \), where \( k \) is the constant of variation.
  
  Rearranging the given equation:
  \[
  3y = -4x
  \]
  \[
  y = -\frac{4}{3}x
  \]
  Since this is in the form \( y = kx \), \( y \) varies directly with \( x \), and the constant of variation is \( k = -\frac{4}{3} \).

### 7. \( 4x - 2y = 1 \)
- Rearrange the equation:
  \[
  2y = 4x - 1
  \]
  \[
  y = 2x - \frac{1}{2}
  \]
  This equation is not in the form \( y = kx \) because of the constant \( -\frac{1}{2} \). Therefore, \( y \) does not vary directly with \( x \).

### 8. If \( y = 13 \) when \( x = 4 \), find \( x \) when \( y = 26 \).
- If \( y \) varies directly with \( x \), then \( y = kx \).
  Using \( y = 13 \) and \( x = 4 \):
  \[
  13 = k \cdot 4 \quad \Rightarrow \quad k = \frac{13}{4}
  \]
  Now, using \( y = 26 \):
  \[
  26 = \frac{13}{4}x
  \]
  Solving for \( x \):
  \[
  x = \frac{26 \times 4}{13} = 8
  \]
  So, \( x = 8 \) when \( y = 26 \).

### 9. If \( y = \frac{3}{4} \) when \( x = \frac{1}{2} \), find \( y \) when \( x = 6 \).
- Use \( y = kx \) again.
  Using \( y = \frac{3}{4} \) and \( x = \frac{1}{2} \):
  \[
  \frac{3}{4} = k \cdot \frac{1}{2} \quad \Rightarrow \quad k = \frac{3}{4} \times 2 = \frac{3}{2}
  \]
  Now, find \( y \) when \( x = 6 \):
  \[
  y = \frac{3}{2} \times 6 = 9
  \]
  So, \( y = 9 \) when \( x = 6 \).

### 10. Find the slope of the line that passes through the points \( (7, 1) \) and \( (-3, -4) \).
- The slope \( m \) is given by the formula:
  \[
  m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 1}{-3 - 7} = \frac{-5}{-10} = \frac{1}{2}
  \]
  So, the slope is \( \frac{1}{2} \).

### 11. Find the slope and \( y \)-intercept of the line \( -3x + 4y = -12 \).
- First, solve for \( y \):
  \[
  4y = 3x - 12
  \]
  \[
  y = \frac{3}{4}x - 3
  \]
  So, the slope is \( \frac{3}{4} \) and the \( y \)-intercept is \( -3 \).

### 12. \( x = -3 \)
- This is a vertical line. The slope of a vertical line is undefined, and it has no \( y \)-intercept, as it is parallel to the \( y \)-axis.

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### Related Questions:
1. How can you determine the constant of variation for other types of relationships?
2. What are the steps to convert a linear equation into slope-intercept form?
3. How do you calculate the slope of a horizontal line?
4. What is the significance of the \( y \)-intercept in a linear equation?
5. How do direct and inverse variation differ in algebra?

### Tip:
For any linear equation, rearranging it into the form \( y = mx + b \) can help you quickly identify the slope \( m \) and the \( y \)-intercept \( b \).

Determine whether y varies directly with x. If so, find the constant of variation. Solve problems related to finding slopes and y-intercepts of linear equations.

Explore how to determine if y varies directly with x, and solve for the constant of variation. Learn to find the slope and y-intercept of lines, and solve various algebraic problems. This content is suitable for students in Grades 7-10.

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