If the average of a and b is 10 and a^2 - b^2 = 80, what is the value of b?
To find the value of b, we need to solve the given equations simultaneously. Let's break down the problem and solve it step by step:
Step 1: Average of a and b is 10.
The average of two numbers is their sum divided by 2. So, we can express this as:
(a + b)/2 = 10
Step 2: Solve equation (a + b)/2 = 10 for a.
Multiply both sides of the equation by 2 to get rid of the fraction:
a + b = 20
Rearrange this equation to isolate a:
a = 20 - b
Step 3: Substitute the value of a in the equation a^2 - b^2 = 80.
Using the expression we found for a in step 2, we can substitute it into the equation:
(20 - b)^2 - b^2 = 80
Step 4: Expand and simplify the equation.
Expanding (20 - b)^2 gives us:
(20 - b)(20 - b) - b^2 = 80
(400 - 40b - 40b + b^2) - b^2 = 80
400 - 80b + b^2 - b^2 = 80
400 - 80b = 80
Step 5: Solve for b.
Subtract 400 from both sides of the equation:
-80b = 80 - 400
-80b = -320
Divide both sides by -80 to isolate b:
b = (-320)/(-80) = 4
Therefore, the value of b is 4.