Problem 85 - Introductory Algebra for College Students

Question

Use the given information to write an equation. Let \(x\) represent the number described in each exercise. Then solve the equation and find the number. When 3 is subtracted from three-fourths of a number, the result is equal to one-half of the number. What is the number?

Step-by-Step Solution

Answer

The number is 12.

Step 1: Translate the problem into an equation

Firstly, let's translate the problem into an equation. The problem states that 'when 3 is subtracted from three-fourths of a number (which we will represent by \(x\)), the result is equal to one-half of the number.'This can be translated to the equation: \(\frac{3}{4}x - 3 = \frac{1}{2}x\)

Step 2: Solve the equation

Start solving this equation by isolating \(x\) on one side. To do this, subtract \(\(\frac{1}{2}x\) from both sides. This gives us: \(\frac{3}{4}x - \frac{1}{2}x = 3\). Simplify it to: \(\(\frac{1}{4}x = 3\)

Step 3: Solve for \(x\)

Finally, to find the number represented by \(x\), multiply both sides by 4. Thus, \(x = 3 \times 4 = 12\). Hence, the number is 12.

Key Concepts

Linear Equations

Linear equations are the backbone of algebra and are crucial in solving many real-world problems. They have one or two variables, usually denoted by letters such as \(x\) or \(y\). The highest power of the variable(s) in a linear equation is always one, which makes the equation "linear". This zero-degree power ensures that solutions manifest as straight lines when graphed.

Linear equations can be represented in the form \(ax + b = 0\), where \(a\) and \(b\) are constants. Solving these equations involves isolating the variable to one side of the equation to find its value.

In our example, the equation \(\frac{3}{4}x - 3 = \frac{1}{2}x\) is a linear one. Here, \(x\) represents the number we need to find, and our task is to simplify and solve the equation to discover its value. Notice how simplifying the equation step-by-step helps to keep the work clear and organized, making it easier to identify the solution.

Problem-solving

Problem-solving is an essential skill used across various disciplines. In mathematics, it involves understanding a situation and finding a methodical approach towards a solution.

When solving linear equations, problem-solving primarily includes:

• Identifying key elements of the given problem or word problem.
• Translating these elements into mathematical expressions.
• Performing calculations in a step-by-step manner to find a solution.

In the given exercise, problem-solving began with identifying what was being asked ("What is the number?"). Next, it involved setting up an equation and solving it logically. Developing a strategy for solving the equation, such as isolating \(x\) and simplifying both sides, ensured that the solution was found effectively and efficiently.

Translating Words into Math

Translating words into math is a crucial step in solving many algebraic problems. When you encounter a problem presented in prose, the goal is to extract the mathematical operations hidden in the words.

Key phrases often indicate certain operations:

• "Is equal to" translates to \(=\).
• "Three-fourths of a number" can be written as \(\frac{3}{4}x\).
• "Subtracted from" suggests a subtraction operation.

In our example, the statement "when 3 is subtracted from three-fourths of a number, the result is equal to one-half of the number" translates to \(\frac{3}{4}x - 3 = \frac{1}{2}x\). Learning to correctly translate these statements into equations is essential for setting up the problem to solve correctly. Utilizing this skill helps bridge the gap between understanding a word problem and executing the mathematical operations to find a solution.