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Chapter 5: Problem 47

Solve. $$ \frac{7}{\frac{1}{4}}=\frac{28}{x} $$

Short Answer

Expert verified
x = 1

Step by step solution

01

Simplify the Left Side of the Equation

Simplify the fraction on the left side of the equation. To do this, divide 7 by \(\frac{1}{4}\). This is equivalent to multiplying 7 by the reciprocal of \(\frac{1}{4}\), which is 4.
02

Perform the Multiplication

Calculate the multiplication: \(7 \times 4 = 28\). So, the equation now simplifies to \(28 = \frac{28}{x}\).
03

Isolate the Variable

Now, isolate \(x\) by recognizing that \(\frac{28}{x} = 28\). To solve for \(x\), note that \(28\) divided by \(28\) equals 1, so \(x = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving equations
Solving equations is a fundamental concept in algebra. It's all about finding the unknown value, often represented by a variable like \(x\). The goal is to rearrange the equation to isolate the variable on one side. In our example, the equation is \(\frac{7}{\frac{1}{4}}=\frac{28}{x}\). The first step is to simplify each side, making it easier to isolate the variable. Understanding how to manipulate and balance equations helps in solving them efficiently.

Key steps include:
  • Simplifying fractions or expressions on each side
  • Using operations like addition, subtraction, multiplication, or division
  • Ensuring operations are performed on both sides to keep the equation balanced
reciprocal
A reciprocal of a number is another number that, when multiplied together, results in 1. For example, the reciprocal of \(\frac{1}{4}\) is 4 because \(\frac{1}{4} \times 4 = 1\). In the original exercise, we use reciprocals to simplify the equation.

To divide by a fraction, it's more straightforward to multiply by its reciprocal. This is why instead of dividing 7 by \(\frac{1}{4}\), we multiply 7 by 4. Mathematically, this looks like this:

\[\frac{7}{\frac{1}{4}} = 7 \times 4\]

Understanding reciprocals is also useful in simplifying and solving more complex algebraic equations.
isolate variable
Isolating the variable means manipulating the equation so that the variable stands alone on one side. In our example, we want to solve for \(x\) in \(\frac{28}{x}=28\).

To isolate \(x\), we need to consider how division and multiplication work. By recognizing that \(\frac{28}{x} = 28\), we see that dividing 28 by \(x\) should give us 28. Therefore, \(x\) must be 1 since \(28 \times 1 = 28\).

Key strategies to isolate variables include:
  • Performing inverse operations
  • Keeping the equation balanced by doing the same operation on both sides

Mastering this skill is essential for solving algebraic equations and understanding more complex mathematical concepts.

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