Problem 51 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

Answer: The equation has one solution.

Step 1: Write down the given function and equation

We are given the function: $$ f(t) = 2t + 7 $$ And we need to find the solutions for the equation: $$ f(t) = 7 $$

Step 2: Substitute function's expression into the equation

Replace f(t) in the equation with the expression of the function: $$ 2t + 7 = 7 $$

Step 3: Solve the equation for t

Now we need to isolate t by subtracting 7 from both sides of the equation: $$ 2t = 0 $$ Then, divide both sides by 2: $$ t = 0 $$

Step 4: Analyze the result

Since we found a unique value for t (t = 0), it means that the given equation f(t) = 7 has only one solution. Therefore, the equation has one solution.

Key Concepts

Functions and Their Role in Mathematics

Functions play a crucial role in mathematics, serving as a way to map every input to an output. A function is usually represented as a formula that describes the relationship between variables. In general, if you have a function like \( f(t) = 2t + 7 \), it tells you how to calculate the output value for any given input \( t \). These outputs are commonly referred to as the image of the input under the function. For the specific example, inserting different values of \( t \) will produce different outputs. However, sometimes special conditions are examined, such as when the function equals a particular number, like \( f(t) = 7 \). These conditions lead us directly into solving equations, where finding which inputs meet these conditions is key.

Solving Equations

Solving equations involves finding which values satisfy the given equation. When you are given \( f(t) = 2t + 7 \) and \( f(t) = 7 \), it's the same as asking: "For which value of \( t \) is \( 2t + 7 \) equal to 7?"This type of problem-solving can include several steps:

• Identify the expression of the function or equation.
• Substitute variables with given values or expressions, if needed.
• Use algebraic manipulations to isolate the variables.
• Solve for the unknowns.

In our problem, we first simplify: substitute \( f(t) \) with \( 2t + 7 \) and then solve \( 2t + 7 = 7 \). Subtract 7 from both sides leading to \( 2t = 0 \). Finally, divide by 2 to find that \( t = 0 \). This straightforward set of operations reveals one solution to the equation.

Finding a Unique Solution

A unique solution means there is exactly one value that satisfies the equation, leaving no ambiguity. In our example, with the equation \( 2t + 7 = 7 \), the solution process showed us that \( t = 0 \).Once we simplify and solve equations, analyzing our results can confirm the type of solutions we encounter:

• A unique solution - only one possible answer, as in this case.
• No solution - indicated when no values satisfy the equation, often seen in contradictions.
• Infinite solutions - any value satisfies the equation, typically seen in identities.

The step-by-step approach we used led us clearly to the unique solution. Recognizing and correctly identifying these solutions is essential in understanding the behavior of functions and their equations.