Let's work through the questions shown in the image:

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.

Would you like more details on any of these questions?

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 $b$.