Question

State whether or not the given equation is linear. $$2^{x}+2^{y}=16$$

Short answer

No, the equation is not linear because it involves exponential terms.

Step-by-step solution

Understanding the Equation

The given equation is \(2^x + 2^y = 16\). To determine if it's linear, recall that an equation is linear if it can be written in the form \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are constants.

Checking for Linear Form

In a linear equation, variables appear only to the first power and are not inside any functions like exponentials. Here, \(x\) and \(y\) are in the exponents, which means they involve exponential functions \(2^x\) and \(2^y\). This form does not match the linear equation form \(ax + by + c = 0\).

Drawing Conclusion

Since \(x\) and \(y\) are exponents, which violate the condition for a linear equation to have variables to the power of 1, the equation \(2^x + 2^y = 16\) is not a linear equation.

Key concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. They take the general form of \(a^x\), where \(a\) is a constant and \(x\) is the variable. In these functions, the variable is not simply multiplied or added; instead, it determines how many times the base number is multiplied by itself. This behavior results in exponential growth or decay, depending on whether the base \(a\) is greater or less than one. Exponential functions can increase or decrease very quickly, making them distinct from linear functions.

• ** Growth**: When the base is greater than 1, such as \(2^x\), the function grows rapidly.
• **Decay**: When the base is between 0 and 1, like \((\frac{1}{2})^x\), the function diminishes swiftly.

In the equation \(2^x + 2^y = 16\), both \(x\) and \(y\) are exponents, showcasing the exponential nature of the equation.

Equation Forms

Equations come in different forms, each with distinct characteristics. Linear equations are a specific type that have variables only to the first power, appearing in the form \(ax + by + c = 0\). This means they do not have exponents, roots, or other nonlinear operations applied to the variables. Instead, the variables are straightforwardly scaled and shifted.

• **Standard form**: Where the coefficients of \(x\) and \(y\) define the line's slope and intercept.
• **Exponential presence**: If an equation involves terms like \(a^x\), it doesn't fit the linear model.

Our original equation, \(2^x + 2^y = 16\), does not fit the linear form due to the presence of exponential terms. This deviation signifies that it is non-linear.

Mathematical Solutions

In mathematics, finding solutions to a problem often requires determining the type of equation at hand. Recognizing whether an equation is linear or nonlinear guides the solving approach and techniques. Linear equations can be solved using simple algebraic methods like substitution or elimination.
Exponential equations, on the other hand, may require logarithms or different strategies to isolate and solve for the variable. For instance, converting an exponential equation into a form involving logarithms can help in finding the solutions.

• **Linear solutions**: Direct and rely on straightforward algebraic manipulation.
• **Non-linear solutions**: May involve transformation or alternative methods like graphing or numerical approximation.

In the context of \(2^x + 2^y = 16\), since it’s identified as non-linear, typical linear methods won't apply. Instead, understanding its exponential form is crucial for further exploration.