Solve for X and Y:
2^x=8^1+y And (x+2y)(x-y)=0
To solve for x and y in the given equations, let's break it down step by step:
Equation 1: 2^x = 8^(1+y)
To solve this equation:
Step 1: Rewrite 8 as a power of 2 since both sides of the equation have a base of 2.
2^x = (2^3)^(1+y)
Step 2: Apply the exponent property (a^(b*c) = (a^b)^c). Multiply exponents within the brackets.
2^x = 2^(3(1+y))
Step 3: Apply the power equality property (if a^x = a^y, then x = y). Set the exponents equal.
x = 3(1+y)
Equation 2: (x+2y)(x-y) = 0
To solve this equation:
Step 1: Apply the zero product property (if ab = 0, then either a = 0 or b = 0).
x + 2y = 0 or x - y = 0
For simplicity, let's solve each equation separately:
Using the equation "x + 2y = 0":
Step 1: Solve for x:
x = -2y
Using the equation "x - y = 0":
Step 1: Solve for x:
x = y
Now we have two values for x:
1. x = -2y
2. x = y
To find y, substitute these values in equation 1: x = 3(1+y)
Substituting x = -2y:
-2y = 3(1+y)
Step 1: Distribute 3 to both terms inside the parentheses:
-2y = 3 + 3y
Step 2: Move all y terms to one side:
3y + 2y = 3
Step 3: Combine like terms:
5y = 3
Step 4: Solve for y by dividing both sides by 5:
y = 3/5
Substituting x = y:
x = 3/5
So the solution for x and y is x = 3/5 and y = 3/5.