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Chapter 3: Problem 34

Solve and check each equation. $$\frac{7}{4} r=42$$

Short Answer

Expert verified
We solve the equation \(\frac{7}{4}r = 42\) by multiplying each side by the reciprocal of the fraction \(\frac{7}{4}\), which is \(\frac{4}{7}\). This gives: \[\frac{4}{7} \cdot \frac{7}{4}r = 42 \cdot \frac{4}{7}\] The fractions cancel each other out, and we get \(r = 6 \cdot 4 = 24\). Plugging back into the original equation, we have \(\frac{7}{4} \cdot 24 = 42\), which is true. Therefore, the solution is \(r = 24\).

Step by step solution

01

Write down the given equation

We begin by writing down the given equation: $$\frac{7}{4}r = 42$$
02

Multiply each side by the reciprocal of the fraction

We now multiply each side of the equation by the reciprocal of the fraction \(\frac{7}{4}\). The reciprocal is \(\frac{4}{7}\). $$\frac{4}{7} \cdot \frac{7}{4}r = 42 \cdot \frac{4}{7}$$
03

Simplify and solve

The fractions on the left side cancel each other out, leaving us with: $$r = 42 \cdot \frac{4}{7}$$ Now, \(42\) can be divided by \(7\): $$r = 6 \cdot 4$$ So, $$r = 24$$
04

Check the solution

To check the solution, we substitute \(r = 24\) back into the original equation and see if both sides are equal: $$\frac{7}{4} \cdot 24 = 42$$ This simplifies to: $$42 = 42$$ Since both sides are equal, the solution \(r = 24\) is correct.

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Key Concepts

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

Understanding Algebra in Equations
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, we use letters like \( r \) to represent unknown values or variables. Solving an equation means finding the value of the variable that makes the equation true.

For example, in the equation \( \frac{7}{4} r = 42 \), the goal is to find the value of \( r \). To achieve this, we isolate \( r \) on one side of the equation. Algebra provides a systematic approach to handling equations and solving for unknowns. It involves performing operations such as addition, subtraction, multiplication, and division in a logical sequence to maintain the equality.
The Concept of Reciprocals
A reciprocal is what you multiply a number by to get one. If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). Multiplying a number by its reciprocal always results in 1.

In solving equations, reciprocals are very useful, particularly when dealing with fractions. In our example, \( \frac{7}{4} \) is multiplied by its reciprocal \( \frac{4}{7} \) to help isolate the variable \( r \). This step simplifies the equation and helps to main the balance of the equation on both sides without changing its equality.
  • Reciprocal of \( \frac{7}{4} \) is \( \frac{4}{7} \)
  • Used to eliminate fractions and simplify equations
  • Ensures each step of the solution keeps the equation balanced
Importance of Checking Solutions
Checking solutions is a critical step in solving equations. This is done to verify that the value we found for \( r \) truly satisfies the original equation.

After solving for \( r \), substitute the value back into the original equation to ensure that both sides of the equation are equal. For instance, by substituting \( r = 24 \) in the original equation \( \frac{7}{4} \cdot 24 = 42 \), both sides become equal, confirming the solution is correct.

This step acts as a confirmation:
  • If both sides equal after substitution, the solution is correct
  • If not, re-check calculations for possible errors
Consistency in outputs demonstrates the reliability of our solution.
Working with Fractions
Fractions represent parts of a whole and are used frequently in algebraic equations. A fraction is composed of a numerator (top number) and a denominator (bottom number).

When solving equations involving fractions, it is often helpful to eliminate them by using techniques such as multiplying by reciprocals. This method simplifies calculations and makes solving easier. Consider:
  • Fractions can express relationships between quantities
  • Understanding fractions is crucial for algebraic manipulation
  • Using reciprocals aids in simplifying fractional equations
Mastery over fractions is essential in solving equations efficiently and accurately.

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