Suppose that \( y \) varies inversely with \( x \), and \( y=6 \) when \( x=4 \). (a) Write an inverse variation equation that relates \( x \) and \( y \). Equation: (b) Find \( y \) when \( x=20 \). \( y=\square \)

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 \( y = 6 \) when \( x = 4 \). We can substitute these values into the equation to find \( k \): \[ 6 = \frac{k}{4} \] To solve for \( k \), we multiply both sides by 4: \[ k = 6 \times 4 = 24 \] ### Step 3: Write the complete inverse variation equation Now that we have \( k \), we can write the complete equation: \[ y = \frac{24}{x} \] ### Step 4: Find \( y \) when \( x = 20 \) Now we need to find \( y \) when \( x = 20 \): \[ y = \frac{24}{20} \] Calculating this gives: \[ y = \frac{24}{20} = \frac{6}{5} = 1.2 \] ### Final Answers (a) The inverse variation equation is: \[ y = \frac{24}{x} \] (b) When \( x = 20 \), \( y = 1.2 \). Thus, the final answer is: \[ y = 1.2 \]