Problem 56 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

Answer: The equation has no solution.

Step 1: Substitute the function into the equation

We're given the equation \(f(t) + 1 = f(t + 1)\). The function is \(f(t) = 2t + 7\). So, let's substitute \(2t + 7\) into the equation for \(f(t)\): $$(2t + 7) + 1 = 2(t+1) + 7$$

Step 2: Simplify and solve for t

Now we need to simplify the equation and solve for \(t\): \begin{align*} (2t + 7) + 1 &= 2(t+1) + 7\\ 2t + 8 &= 2t + 2 + 7\\ 2t + 8 &= 2t + 9 \end{align*}

Step 3: Analyze if there is no solution, one solution, or an infinite number of solutions

Now let's determine the number of solutions to the equation: If we subtract \(2t\) from both sides of the equation, we get: $$8 \neq 9$$ The result is a statement, not an equation involving \(t\). This means there's no value of \(t\) that can make the original equation true. Therefore, the given equation has no solution.

Key Concepts

Linear Functions

A linear function is one of the simplest and most commonly encountered types of functions in algebra. Its general form is defined as \( f(x) = mx + b \), where \( m \) and \( b \) are constants. Here, \( m \) represents the slope, indicating how steep the line is, while \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.
In our exercise, we deal with the linear function \( f(t) = 2t + 7 \). This tells us that the line has a slope of 2, meaning it rises 2 units vertically for every 1 unit it moves horizontally to the right. The y-intercept is 7, meaning the line crosses the y-axis at 7.
Linear functions are important because they depict constant rates of change. They are used in various applications such as calculating speed, predicting profits, and even in everyday decision-making processes. Understanding their fundamental properties can greatly enhance your problem-solving skills.

Equation Solving

Equation solving is a fundamental aspect of algebra. It involves finding the value of the variable that makes an equation true. The process typically requires simplifying the equation, isolating the variable, and then solving for it.
In the original exercise, we are asked to solve the equation \( f(t) + 1 = f(t + 1) \) where \( f(t) = 2t + 7 \). To solve it:

• First, substitute \( f(t) \) with \( 2t + 7 \), simplifying both sides.
• Next, equate and simplify to \( 2t + 8 = 2t + 9 \).
• Finally, on analyzing, the variable cancels and we are left with a statement, \( 8 eq 9 \), showing a contradiction.

The process demonstrates that effective equation solving often involves recognizing when an equation reaches a statement that doesn't involve variables anymore. It helps affirm the type of solution or inconsistency within an equation.

Number of Solutions

When dealing with equations, determining the number of solutions is crucial. It tells us whether an equation can be satisfied and by how many values.
There are typically three scenarios:

• No Solution: This occurs when simplifying leads to a false statement, such as \( 8 eq 9 \). No possible value will satisfy the equation.
• One Solution: This is the most common scenario, where simplifying leads to a true statement with one definite solution, such as \( x = 3 \).
• Infinite Solutions: This happens when simplifying results in a trivial truth, like \( 0 = 0 \), valid for any value of the variable.

In our exercise, the outcome \( 8 eq 9 \) clearly indicates no solution, as there's no \( t \) value that will satisfy the original equation. Recognizing these types of solutions helps in validating results and ensuring the logic of algebraic manipulations.