Suppose that $$ y $$ varies inversely, with $$ x $$, and $$ y=5 $$ when $$ x=4 $$. (a) Write an inverse variation equation that relates $$ x $$ and $$ y $$. Equation: (b) Find $$ y $$ when $$ x=10 $$. $$ y=\square $$

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To solve the problem, we will follow the steps outlined:

### Step 1: Write the inverse variation equation

Since $$ y $$ varies inversely with $$ x $$, we can express this relationship as:

$$
y = \frac{k}{x}
$$

where $$ k $$ is a constant.

### Step 2: Find the constant $$ k $$

We know that $$ y = 5 $$ when $$ x = 4 $$. We can substitute these values into the equation to find $$ k $$:

$$
5 = \frac{k}{4}
$$

To solve for $$ k $$, we multiply both sides by 4:

$$
k = 5 \times 4 = 20
$$

### Step 3: Write the complete inverse variation equation

Now that we have $$ k $$, we can write the complete equation:

$$
y = \frac{20}{x}
$$

### Step 4: Find $$ y $$ when $$ x = 10 $$

Now we need to find $$ y $$ when $$ x = 10 $$:

$$
y = \frac{20}{10}
$$

Calculating this gives:

$$
y = 2
$$

### Final Answers

(a) The inverse variation equation is:

$$
y = \frac{20}{x}
$$

(b) When $$ x = 10 $$, $$ y = 2 $$.

Thus, the final answer is:

$$
y = 2
$$
(a) The inverse variation equation is $$ y = \frac{20}{x} $$. (b) When $$ x = 10 $$, $$ y = 2 $$.

Suppose that \( y \) varies inversely, with \( x \), and \( y=5 \) when \( x=4 \). (a) Write an inverse variation equation that relates \( x \) and \( y \). Equation: (b) Find \( y \) when \( x=10 \). \( y=\square \)

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