Problem 53 - Algebra Form and Function

Question

\(f(t)=2 t+7\). Does the equation have no solution, one solution, or an infinite number of solutions? $$ f(t)=f(t+1)-2 $$

Step-by-Step Solution

Answer

Answer: The given equation has an infinite number of solutions.

Step 1: Write down the given equation

The given equation is \(f(t) = f(t+1) - 2\). We first need to substitute the function \(f(t) = 2t + 7\) into both sides of the equation.

Step 2: Substitute the function into the equation

We substitute \(f(t)=2t+7\) and \(f(t+1)=2(t+1)+7\) into the given equation as follows: $$ 2t+7=2(t+1)+7−2 $$

Step 3: Simplify the equation

We now simplify the equation to make it easier to solve for the variable \(t\): $$ 2t+7=2t+2+7−2 $$ $$ 2t+7=2t+7 $$

Step 4: Solve for t

Once we try to solve for \(t\), we'll be able to determine the number of solutions. $$ 2t+7=2t+7 $$ Subtracting \(2t\) from both sides, we get: $$ 7=7 $$

Step 5: Determine the number of solutions

Since the equation \(7=7\) is always true regardless of the value of \(t\), the given equation has an infinite number of solutions.

Key Concepts

Infinite Solutions

When we say an equation has infinite solutions, it means that there is not just one specific value that satisfies the equation. Instead, any value you choose will make the equation true. In the solution given, we found that after simplifying the equation, we end up with the statement \(7 = 7\). This is an identity, a statement that is always true, telling us that the equation does not "care" which value you choose for \(t\); the equality holds no matter what.This happens in linear equations when, after simplification, all terms involving the variables cancel out and we are left with a true statement involving only constants. Such characteristics hint that the given equation is not just balanced for one particular variable value, but for an *infinite array* of values instead.Next time you're faced with a similar situation, remember that equations like this - where any value satisfies them - offer the intriguing concept of limitless possibilities within defined constraints!

Equation Simplification

Simplifying equations is like peeling layers off an onion, unveiling the simple truths hiding underneath complex exteriors. Simplification involves combining like terms, distributing numbers, and canceling terms on both sides of the equation to trim it down to its simplest form.In our original problem, after substituting the known function \(f(t)=2t+7\) into the equation \(f(t) = f(t+1) - 2\), we simplify to:\[2t+7 = 2(t+1) + 7 - 2\]Breaking it down further results in:- Distributing on the right side to get \(2t + 2\)- Simplifying constants gives us \(7\), arriving at our simple equation \(2t + 7 = 2t + 7\)During simplification, it’s usually a good idea to double-check each step to avoid any metaphoric tears from redundant or unintentional complexity, ensuring clarity and ultimately arriving at the solution or identity more efficiently!

Problem Solving

Tackling problems like these can be approached methodically by following a structured path:

• **Understand the problem**: Start by clearly defining what's given and what's sought - a step often helped by re-wording the equation or visualizing it.
• **Substitute accordingly**: Insert known functions or constants into the equation. This embeds familiar elements into the unknown, reducing the complexity.
• **Simplify systematically**: Carefully eliminate or adjust variables and constants through arithmetic operations to distill the core equation.
• **Analyze and conclude**: When simplified, examine what's left; in our problem, the comparison resulted in an identity like \(7 = 7\) pointing to infinite solutions.

This step-by-step methodology in problem-solving ensures that solutions are reached with minimal errors, turning potentially intricate puzzles into approachable feats.