Problem 56 - Algebra A Combined Function

Question

Write each algebraic expression described. A 5 -foot piece of string is cut into two pieces. If one piece is \(x\) feet long, express the other length in terms of \(x\).

Step-by-Step Solution

Answer

The other piece is \(5 - x\) feet long.

Step 1: Understanding the Problem

You have a piece of string that is 5 feet long. It’s cut into two pieces, and one of the pieces is labeled as \(x\) feet long. Your task is to find the length of the other piece using algebra.

Step 2: Set Up the Equation

The equation that represents the total length of the string is the sum of the lengths of the two pieces. In algebraic terms, it can be written as:\[x + ext{other piece} = 5\]

Step 3: Express the Other Piece

To find the length of the other piece, we need to isolate it in the equation. Subtract \(x\) from both sides of the equation:\[\text{other piece} = 5 - x\]

Step 4: Final Expression

The length of the other piece of string, in terms of \(x\), is given by the expression \(5 - x\). This expression shows how much is left of the total 5 feet after cutting off \(x\) feet.

Key Concepts

Variables in Algebra

Variables in algebra are symbols that represent unknown values. Typically, letters such as \(x\), \(y\), or \(z\) are used. In our exercise with the string, \(x\) represents the length of one piece of the string after it is cut.

• Variables can vary, meaning they can take various values depending on the situation.
• They allow us to create general mathematical models, which is helpful in problem-solving.

In algebra, using variables enables us to express relationships in a formulaic way. For example, the original problem with the string translates into the equation \(x + \text{other piece} = 5\). Here, \(x\) represents the length of one piece of string, helping us express the relationship between the known total length and the unknown piece.

Linear Equations

A linear equation is an algebraic statement involving variables where each term is either a constant or the product of a constant and a single variable. The equation from our problem is a simple linear equation: \[x + \text{other piece} = 5\]This equation is linear because the variables are not raised to a power other than one.

• Linear equations are so-named because their graph is a straight line.
• They often appear in real-life situations, such as calculating distances or budgets.

In our scenario, solving the equation involves finding the value of the 'other piece' of string. By following simple algebraic steps, you can quickly find that the other piece equals \(5 - x\). Linear equations help translate word problems into mathematical statements we can solve.

Substitution Method

The substitution method is a straightforward technique used to solve systems of equations and also useful in breaking down expressions.

• It involves replacing a variable with its equivalent expression.
• This method simplifies and clarifies what we are trying to find.

In our problem, once we identify \(x\) as the length of one piece, the substitution method allows us to find the length of the other piece by substituting into our equation. After rearranging the original equation \(x + \text{other piece} = 5\), we can subtract \(x\) from each side to directly substitute the expression: \[\text{other piece} = 5 - x\]This gives us a clear understanding of how much string remains after removing the piece that is \(x\) feet long. The simplicity and efficiency of the substitution method make it a favorite tool among mathematicians and students alike.