Question

the value of y varies directly with x. when the value of x is 4, the value of y is -12. what is the constant of variation when y is a function of x, and what is the value of y when x = -6? move the correct answer to each box. each answer may be used more than once. not all answers will be used. -18 -3 -2 -1/3 1/3 2 3 18 the constant of variation is \boxed{}. the value of y when x = -6 is \boxed{}.

Explanation:

Step1: Recall direct variation formula

For direct variation, the formula is \( y = kx \), where \( k \) is the constant of variation. We know that when \( x = 4 \), \( y=- 12 \). Substitute these values into the formula to find \( k \).
\( -12=k\times4 \)

Step2: Solve for \( k \)

To find \( k \), divide both sides of the equation \( -12 = 4k \) by 4.
\( k=\frac{-12}{4}=- 3 \)
So the constant of variation \( k \) is \( -3 \).

Step3: Find \( y \) when \( x = - 6 \)

Now that we know \( k=-3 \), the equation of variation is \( y=-3x \). Substitute \( x = - 6 \) into this equation.
\( y=-3\times(-6) \)

Step4: Calculate \( y \)

Multiply \( -3 \) and \( -6 \). \( -3\times(-6)=18 \)? Wait, no, wait: Wait, \( k=-3 \), so \( y = kx=-3x \). When \( x=-6 \), \( y=-3\times(-6)=18 \)? Wait, no, wait, let's check again. Wait, when \( x = 4 \), \( y=-12 \), so \( k=\frac{y}{x}=\frac{-12}{4}=-3 \). Then the equation is \( y=-3x \). So when \( x=-6 \), \( y=-3\times(-6)=18 \)? Wait, but let's check the options. Wait, the options include 18. Wait, but let's re - check the calculation. \( -3\times(-6)=18 \), yes. But wait, maybe I made a mistake? Wait, no, direct variation: \( y = kx \), \( k=\frac{y}{x}=\frac{-12}{4}=-3 \). Then for \( x=-6 \), \( y=-3\times(-6)=18 \).

Wait, but let's check the first part again. The constant of variation is \( k=-3 \), which is one of the options. Then the value of \( y \) when \( x = - 6 \) is \( 18 \), which is also one of the options.

Wait, but let's re - do the steps:

1. Direct variation formula: \( y=kx \), \( k \) is constant.
2. Given \( x = 4 \), \( y=-12 \). Substitute into \( y = kx \): \( -12=k\times4\Rightarrow k=\frac{-12}{4}=-3 \). So constant of variation is \( -3 \).
3. Now, to find \( y \) when \( x=-6 \), use \( y = kx=-3\times(-6)=18 \).

Answer:

The constant of variation is \(-3\).
The value of \( y \) when \( x=-6 \) is \( 18 \).