\( K \) Find an equation of the inverse relation. \( y=5 x+7 \) A. \( y=\frac{x}{5}-7 \) B. \( x=5 y-7 \) O. \( x=5 y+7 \) OD. \( x=\frac{y+7}{5} \)

To find the inverse of the given equation \( y = 5x + 7 \), we start by swapping \( x \) and \( y \) to solve for \( y \). This gives us \( x = 5y + 7 \). Now, isolate \( y \) by subtracting 7 from both sides: \( x - 7 = 5y \). Finally, divide by 5 to get \( y = \frac{x - 7}{5} \). Therefore, in terms of the options provided, we rearrange it as \( x = \frac{y + 7}{5} \), which matches option OD. So the correct answer is OD: \( x = \frac{y + 7}{5} \). One common mistake when finding inverses is forgetting to swap \( x \) and \( y \) first. Many students will just solve for \( y \) without that step, leading to incorrect results. Always remember: swap first, then solve! If you're curious to dive deeper into inverses, the concept of functions being one-to-one is crucial. A function needs to pass the horizontal line test to ensure it has an inverse. If a horizontal line crosses the function more than once, it means there's no unique inverse relation for that function — a fun twist in the world of algebra!