Sure, let's solve the two problems step-by-step:
Problem 1: If y varies inversely as x, and y=9 when x=2, find y when x=3.
Step 1: When two variables are inversely proportional, the product of the two variables is constant. Hence, their product when y=9 and x=2 should equal their product when y is some unknown value we'll call y2 and x=3. This idea can be represented by the equation y1*x1 = y2*x2.
Step 2: Substitute the known values into the equation. This gives us: 9*2 = y2*3, which simplifies to 18 = 3*y2.
Step 3: Divide both sides of the equation by 3 to isolate y2 on one side of the equation. Hence, y2 = 18 / 3 = 6.0.
So, if y varies inversely as x, and y=9 when x=2, then y=6.0 when x=3.
Problem 2: If x and y vary inversely as each other, and x=10 when y=6. Find y when x=15.
Step 1: Similar to problem 1, since x and y vary inversely, then the product of x1 and y1 equals the product of x2 and some unknown value of y that we'll call y2. Again, this can be represented by the equation: x1*y1 = x2*y2.
Step 2: Substitute known values into the equation. So, we have: 10*6 = 15*y2, which simplify to 60 = 15 * y2.
Step 3: Divide both sides of the equation by 15 to isolate y2. Thus, we have y2 = 60 / 15 = 4.0.
So, if x and y vary inversely as each other, and x=10 when y=6, then y=4.0 when x=15.