Solve: 2^{x+1} * 2^{3y+1} = 8 and 2^{x+2} * 2^{2y+2} = 16

Steps to Solve

1. Rewrite the equations using properties of exponents

Since $8 = 2^3$ and $16 = 2^4$, we can rewrite the equations as:

• The first equation: $$ 2^{x+1} \cdot 2^{3y+1} = 2^3 $$
• The second equation: $$ 2^{x+2} \cdot 2^{2y+2} = 2^4 $$

2. Combine the exponents

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can combine the left-hand sides:

• For the first equation: $$ 2^{(x+1)+(3y+1)} = 2^3 $$ This simplifies to: $$ 2^{x + 3y + 2} = 2^3 $$

• For the second equation: $$ 2^{(x+2)+(2y+2)} = 2^4 $$ This simplifies to: $$ 2^{x + 2y + 4} = 2^4 $$

3. Set the exponents equal

Since the bases are equal, we can set the exponents equal to each other:

• From the first equation: $$ x + 3y + 2 = 3 $$
• From the second equation: $$ x + 2y + 4 = 4 $$

4. Solve for one variable from one equation

From the first equation, isolate $x$: $$ x = 3 - 3y - 2 $$ $$ x = 1 - 3y $$

5. Substitute into the second equation

Now substitute for $x$ in the second equation: $$ (1 - 3y) + 2y + 4 = 4 $$

6. Simplify and solve for $y$

Combine like terms: $$ 1 - 3y + 2y + 4 = 4 $$ This simplifies to: $$ 5 - y = 4 $$ Isolating $y$ gives: $$ -y = 4 - 5 $$ $$ -y = -1 $$ $$ y = 1 $$

7. Use the value of $y$ to find $x$

Now substitute $y = 1$ back into the expression for $x$: $$ x = 1 - 3(1) $$ $$ x = 1 - 3 $$ $$ x = -2 $$