Problem 48 - Elementary Algebra

Question

Translate each sentence to a mathematical statement and then simplify. The base of a triangle is 3 feet shorter than its height. If the height measures 5 feet, find the length of the base.

Step-by-Step Solution

Answer

The base of the triangle is 2 feet.

Step 1: Identify Variables

First, identify the variables involved in the problem. Let the height of the triangle be denoted by \( h \) and the base of the triangle by \( b \).

Step 2: Assign Known Values

From the problem, we know that the height \( h \) is 5 feet. Therefore, \( h = 5 \).

Step 3: Formulate the Base Equation

According to the problem, 'The base of a triangle is 3 feet shorter than its height.' This can be written as an equation: \( b = h - 3 \).

Step 4: Substitute the Known Value

Substitute the known value of the height into the equation for the base. So, replace \( h \) with 5 in \( b = h - 3 \) to get \( b = 5 - 3 \).

Step 5: Simplify the Expression

Simplify the expression \( b = 5 - 3 \) to find the length of the base. Therefore, \( b = 2 \).

Step 6: Final Step: Conclusion

The length of the base of the triangle is 2 feet.

Key Concepts

Understanding Variables

In algebra, a variable is a symbol, often a letter like \( x \) or \( y \), used to represent an unknown value. It acts as a placeholder that can be replaced with a number. Utilizing variables allows us to construct and solve mathematical expressions and equations efficiently.

For example, in the exercise provided, we have two main variables: \( h \) for the height of the triangle and \( b \) for the base. Variables help simplify complex sentences into manageable parts and give us a way to manipulate and evaluate the situation numerically.

This flexibility is fundamental when it comes to finding unknown values, making variables a cornerstone of algebra.

Formulating Equations

An equation is a mathematical statement where two expressions are equal, shown with the \( = \) sign. Equations are the backbone of solving problems in algebra because they allow us to express relationships between variables.

In the provided solution, the sentence describing the triangle is translated into the equation \( b = h - 3 \).

• The left-hand side \( b \) represents the base of the triangle.
• The right-hand side \( h - 3 \) expresses that the base is 3 feet shorter than the height \( h \).

By setting up this equation, we have formed a direct relationship between the base and height, which we can solve to find specific values.

The Simplification Process

Simplification in algebra involves reducing an expression or equation to its simplest form. This makes it easier to work with and understand.

After creating the equation for the base, \( b = h - 3 \), we simplify by substituting the known value of the height, \( h = 5 \), leading to \( b = 5 - 3 \). This step-by-step simplification helps break down the problem and makes the solution clearer.

By calculating \( b = 2 \), we find the simplest form of the expression, solving the problem efficiently. Simplification can include operations like addition, subtraction, multiplication, and division, all aiming to make equations easier to handle.

Translating Sentences into Equations

Translating sentences into equations is a vital skill in algebra. It involves converting descriptive language into mathematical expressions that capture the essence of a problem.

The sentence "The base of a triangle is 3 feet shorter than its height" translates into the equation \( b = h - 3 \). This conversion is key to solving the exercise.

• "The base of a triangle" becomes the variable \( b \).
• "is 3 feet shorter than" indicates subtraction from another value.
• "its height" substitutes as the variable \( h \).

This translation process is essential as it forms the basis for setting up equations that allow for further analysis and problem-solving in algebra.