Solve the System of Equations 2^(x+y)=16 2^(2x+y)=1/8

Solution:

Step 1: Isolate $x$ in the first equation $2^{x+y}=16$.

• Express both sides of the equation with the same base: $2^{x+y}=2^4$.

• Equate the exponents since the bases are equal: $x+y=4$.

• Rearrange to solve for $x$: $x=4-y$.

Step 2: Substitute $x$ with $4-y$ in the second equation $2^{2x+y}=\frac{1}{8}$.

• Insert $4-y$ in place of $x$: $2^{2(4-y)+y}=\frac{1}{8}$.

• Expand and simplify the exponent:

  • Apply the distributive property: $2^{8-2y+y}=\frac{1}{8}$.

  • Combine like terms: $2^{8-y}=\frac{1}{8}$.

Step 3: Solve for $y$ in the equation $2^{8-y}=\frac{1}{8}$.

• Use the negative exponent rule to rewrite $\frac{1}{8}$: $2^{8-y}=2^{-3}$.

• Since the bases are equal, set the exponents equal to each other: $8-y=-3$.

• Solve for $y$:

  • Isolate $y$: $-y=-3-8$.

  • Simplify: $-y=-11$.

  • Divide by $-1$ to get $y$: $y=11$.

Step 4: Find $x$ by substituting $y=11$ back into $x=4-y$.

• Replace $y$ with $11$: $x=4-11$.

• Simplify to find $x$: $x=-7$.

Step 5: Write the solution as an ordered pair.

• The solution is the pair of values for $x$ and $y$: $(-7,11)$.

Step 6: Present the result in different forms.

• Point Form: $(-7,11)$.

• Equation Form: $x=-7,y=11$.

Knowledge Notes:

To solve a system of equations where the equations are in the form of exponents with the same base, we can use the property that if $a^m=a^n$, then $m=n$ provided that $a$ is not equal to $0$ or $1$. This allows us to equate the exponents and solve for the variables algebraically.

In this problem, we used the following steps:

1. Isolation of a Variable: We first isolated one variable, $x$, in one of the equations.

2. Substitution: We substituted the expression for $x$ into the other equation to solve for $y$.

3. Simplification: We simplified the equations by applying algebraic rules such as the distributive property and combining like terms.

4. Negative Exponent Rule: We used the rule that $a^{-n}=\frac{1}{a^n}$ to rewrite expressions with negative exponents.

5. Solving for the Second Variable: We solved for the second variable, $y$, by equating the exponents with the same base.

6. Back-Substitution: We substituted the value of $y$ back into the expression for $x$ to find its value.

7. Solution Representation: We represented the solution as an ordered pair and in equation form.

These steps are a systematic approach to solving systems of exponential equations and can be applied to similar problems.