In the realm of algebra, one of the most common types of problems involves solving equations for unknown variables. In this article, we will break down the process of solving the equation ( \frac{3}{5} w = 18 ). We will explore the solution stepbystep, provide practical examples, and offer additional context to deepen your understanding.
Understanding the Equation
The equation given is:
[ \frac{3}{5} w = 18 ]
Here, ( w ) represents the unknown variable we need to solve for. The fraction ( \frac{3}{5} ) indicates that ( w ) is being multiplied by this fraction.
StepbyStep Solution
To isolate ( w ), follow these steps:
Step 1: Eliminate the Fraction
To get rid of the fraction, multiply both sides of the equation by the reciprocal of ( \frac{3}{5} ), which is ( \frac{5}{3} ):
[ \frac{5}{3} \cdot \frac{3}{5} w = \frac{5}{3} \cdot 18 ]
On the left side, ( \frac{5}{3} \cdot \frac{3}{5} ) simplifies to ( 1 ):
[ w = \frac{5}{3} \cdot 18 ]
Step 2: Perform the Multiplication
Now, calculate ( \frac{5}{3} \cdot 18 ):

First, divide 18 by 3: [ 18 \div 3 = 6 ]

Then multiply by 5: [ 6 \cdot 5 = 30 ]
Step 3: Write the Solution
Thus, the solution to the equation is:
[ w = 30 ]
Practical Examples
Understanding how to solve this equation can be beneficial in various reallife scenarios. For example:

Budgeting: If you know that ( \frac{3}{5} ) of your monthly income ( w ) corresponds to an expense of $18, you can easily find out your total income.

Distance and Rate Problems: If a certain rate of travel (expressed as a fraction of total distance) leads to a negative distance (like a deficit), you can calculate how far behind you might be.
Additional Analysis
Why Use the Reciprocal?
In algebra, multiplying by the reciprocal is a common technique used to eliminate fractions and simplify equations. It allows us to clear terms efficiently and solve for the variable without changing the balance of the equation.
RealWorld Applications of Linear Equations
Linear equations, such as ( \frac{3}{5} w = 18 ), have wide applications:
 Finance: In financial calculations, being able to express expenses and incomes using linear equations helps in budgeting and forecasting.
 Physics: Equations often describe the relationships between different physical quantities, like distance, time, and speed.
Conclusion
In conclusion, the equation ( \frac{3}{5} w = 18 ) can be solved systematically to find that ( w = 30 ). Understanding how to manipulate equations is not just an academic exercise but a vital skill that can aid in various practical situations. By mastering these concepts, you can improve your problemsolving skills across multiple disciplines.
Attribution
This article utilized foundational concepts adapted from problem discussions available on Brainly by various authors, including user contributions who helped clarify steps in solving equations. Always refer to such platforms for collaborative learning and insight into algebraic concepts.
With this detailed breakdown and additional context, readers can gain a deeper understanding of solving linear equations and their practical relevance.