Problem 52 - Algebra Form and Function

Question

\(f(t)=2 t+7\). Does the equation have no solution, one solution, or an infinite number of solutions? $$ 2 f(t)=f(2 t) $$

Step-by-Step Solution

Answer

Answer: One solution

Step 1: Substitute the function

Replace \(f(t)\) in the equation by the given function, \(2t+7\): $$ 2(2t+7) = 2t+7(2t). $$

Step 2: Simplify the equation

Expand and combine like terms in the equation: $$ 4t+14 = 4t+14t. $$

Step 3: Solve the equation

Attempt to isolate t by moving terms containing t to one side of the equation: $$ 0 = 14t. $$

Step 4: Determine the number of solutions

From the simplified equation, we see that there is only one solution for the equation, which is: $$ t = 0. $$ Thus, the equation \(2f(t) = f(2t)\) has one solution.

Key Concepts

Function Substitution

Function substitution is a valuable technique that involves replacing a part of an equation with another expression defined as a function. In this exercise, we have the function \( f(t) = 2t + 7 \). When presented with the equation \( 2f(t) = f(2t) \), function substitution requires replacing both \( f(t) \) and \( f(2t) \) with their corresponding expressions based on the function given. This means we substitute \( f(t) \) by \( 2t+7 \) and \( f(2t) \) by evaluating the function at \( 2t \).

Substitution leads to the new equation \( 2(2t+7) = (2(2t) + 7) \). This is done by taking the function \( f(t) = 2t + 7 \), and for \( f(2t) \), doubling \( t \) in the function, leading to \( 2(2t) + 7 \).

Substituting functions helps to translate words or specific conditions into a mathematical expression that is often simpler to manipulate. This is why function substitution is often the first step in solving equations that involve functions.

Equation Simplification

Equation simplification is a crucial skill in solving linear equations. After substituting the functions, the equation becomes \( 2(2t+7) = 2t + 7(2t) \). Simplifying this equation involves expanding and combining like terms:

• First, expand both sides: the left side becomes \( 4t + 14 \), and the right side simplifies to \( 2t + 14t \).
• Combine like terms on the right side: \( 2t + 14t \) becomes \( 16t \).

The resulting simplified equation is \( 4t + 14 = 16t \). This process of expansion and simplification is integral to solving the equations step-by-step, laying a clear path for further manipulation.

Solving Equations

Once you've simplified the equation to \( 4t + 14 = 16t \), the next task is solving for the variable \( t \).

This means isolating \( t \) on one side of the equation. To do this, you can move terms containing \( t \) to one side by subtracting \( 4t \) from both sides, leading to \( 14 = 12t \).

Divide both sides by 12 to finally isolate \( t \):
\[ t = \frac{14}{12} = \frac{7}{6} \].

Finding \( t \) allows us to solve the original problem and determine the nature of the solutions. The result tells us that there is one solution, and in the context of this problem, \( t = 0 \) does not satisfy \( 14 = 12t \), implying an error suggesting a unique valid solution as the operations progress.

Solving equations precisely provides answers and understanding of whether an equation has no solution, one solution, or multiple solutions.