22. \( p 1: 5(x-5)+2(y+2)+4(z-1)=0 \), and \( p 2: 3 x-4(y-1)+2(z-1)=0 \)

Let's break down the equations of the planes \( p_1 \) and \( p_2 \) to see how they relate geometrically and explore their implications. The plane \( p_1 \) simplifies to \( 5x + 2y + 4z = 45 \) upon distributing the \( 5 \), \( 2 \), and \( 4 \) and moving everything to one side. On the other hand, the equation for the plane \( p_2 \) simplifies to \( 3x - 4y + 2z = 6 \). When you're trying to find the intersection of these two planes, you essentially have a system of equations to solve. The intersection could potentially be a line, and finding that line can give you insight into the geometric relationship between the two planes. To solve these planes, a common mistake is to forget to arrange the equations correctly or misinterpreting the coefficients while simplifying. Make sure to check your algebra as you manipulate the equations! Often, sketching a simple graph or visualizing the planes in 3D can help you understand their relationship and where they intersect.