Answer:
A = 8 and B = 6
Step-by-step explanation:
Since (B = A - 2), we can substitute (A - 2) in for the "B" variable in the first equation to isolate "A".
(A - 5)² + (B - 3)² = 18 -----> Original equation
(A - 5)² + ((A - 2) - 3)² = 18 -----> Plug in B = A - 2
(A - 5)² + (A - 5)² = 18 -----> Subtract
(A² - 10A + 25) + (A² - 10A + 25) = 18 -----> Expand parentheses
2A² - 20A + 50 = 18 ------> Add like terms
2A² - 20A + 32 = 0 -----> Subtract 18 from both sides
2(A² - 10A + 16) = 0 -----> Remove common factor
2(A - 2)(A - 8) = 0 -----> Factor within parentheses
A = 2 -----> Find A - 2 = 0
A = 8 -----> Find A - 8 = 0
Since "A" gave two possible values, we need to plug them into both equations to see which value gives reasonable "B" values.
When A = 2:
B = A - 2 -----> Original equation
B = 2 - 2 -----> Plug in A = 2
B = 0 -----> Subtract
(A - 5)² + (B - 3)² = 18 -----> Original equation
(2 - 5)² + (B - 3)² = 18 ------> Plug in A = 2
(-3)² + (B - 3)² = 18 -----> Subtract within first parentheses
9 + (B - 3)² = 18 -----> Square value within first parentheses
(B - 3)² = 9 -----> Subtract 9 from both sides
B - 3 = 3 -----> Take square root of both sides
B = 6 -----> Add 3 to both sides
When A = 8:
B = A - 2 -----> Original equation
B = 8 - 2 -----> Plug in A = 8
B = 6 -----> Subtract
(A - 5)² + (B - 3)² = 18 -----> Original equation
(8 - 5)² + (B - 3)² = 18 ------> Plug in A = 8
(3)² + (B - 3)² = 18 -----> Subtract within first parentheses
9 + (B - 3)² = 18 -----> Square value within first parentheses
(B - 3)² = 9 -----> Subtract 9 from both sides
B - 3 = 3 -----> Take square root of both sides
B = 6 -----> Add 3 to both sides
As you can see, when A = 2, there are two possible values of "B" depending on the equation. However, when A = 8, both equations give a "B" value of B = 6. Therefore, A = 8 and B = 6 are the answers.
