Simplifying Algebraic Equations A Step-by-Step Guide
In the realm of mathematics, simplification stands as a cornerstone skill. It allows us to distill complex expressions into their most manageable forms, revealing the underlying structure and relationships with greater clarity. This article delves into the process of simplifying a given equation, highlighting the techniques and steps involved in arriving at a concise and equivalent representation. Specifically, we will be focusing on the equation:
\frac{4x}{2} + \frac{6y}{3} = 12
This equation, while seemingly straightforward, presents an opportunity to showcase the power of simplification. By carefully applying the rules of arithmetic and algebra, we can transform this equation into a more elegant and insightful form.
Understanding the Importance of Simplification
Before we embark on the simplification process, it's crucial to appreciate why simplification is such a fundamental concept in mathematics. Simplified equations are easier to understand, manipulate, and solve. They often reveal hidden patterns and relationships that might be obscured in the original form. Moreover, simplification is an essential step in various mathematical tasks, such as solving systems of equations, graphing functions, and proving theorems.
In the context of the given equation, simplification can lead to a more intuitive understanding of the relationship between x and y. It can also make it easier to identify key features of the equation, such as its slope and intercepts if it represents a line. Ultimately, simplification empowers us to work with mathematical expressions more effectively and efficiently.
Step-by-Step Simplification
Now, let's dive into the process of simplifying the equation:
\frac{4x}{2} + \frac{6y}{3} = 12
The key to simplification lies in performing operations that maintain the equation's equality while reducing its complexity. We will proceed step by step, explaining the rationale behind each operation.
Step 1: Simplify the Fractions
The first step involves simplifying the fractions present in the equation. We observe that both fractions can be reduced by dividing the numerator and denominator by their greatest common divisor.
For the first fraction, 4x/2, the greatest common divisor of 4 and 2 is 2. Dividing both the numerator and denominator by 2, we get:
\frac{4x}{2} = \frac{2 * 2x}{2 * 1} = 2x
Similarly, for the second fraction, 6y/3, the greatest common divisor of 6 and 3 is 3. Dividing both the numerator and denominator by 3, we get:
\frac{6y}{3} = \frac{2 * 3y}{3 * 1} = 2y
Substituting these simplified fractions back into the original equation, we obtain:
2x + 2y = 12
This equation is significantly simpler than the original, as it no longer contains fractions. This simplification makes the equation easier to work with in subsequent steps.
Step 2: Divide by the Common Factor
We now observe that all the terms in the equation 2x + 2y = 12 have a common factor of 2. To further simplify the equation, we can divide both sides of the equation by this common factor.
Dividing both sides by 2, we get:
\frac{2x + 2y}{2} = \frac{12}{2}
This simplifies to:
x + y = 6
This is the simplified form of the original equation. It represents a linear relationship between x and y and is much easier to interpret and manipulate.
Result and Interpretation
After performing the simplification steps, we have transformed the original equation:
\frac{4x}{2} + \frac{6y}{3} = 12
into its simplified form:
x + y = 6
This simplified equation reveals a clear linear relationship between x and y. It tells us that the sum of x and y must always equal 6. This form is much easier to understand and work with than the original equation.
Interpreting the Simplified Equation
The simplified equation x + y = 6 represents a straight line in the Cartesian coordinate system. To understand this line better, we can find its intercepts.
- x-intercept: To find the x-intercept, we set y = 0 and solve for x:
So, the x-intercept is (6, 0).x + 0 = 6 x = 6 - y-intercept: To find the y-intercept, we set x = 0 and solve for y:
So, the y-intercept is (0, 6).0 + y = 6 y = 6
With the intercepts, we can visualize the line represented by the equation. The line passes through the points (6, 0) and (0, 6).
Slope of the Line
We can also determine the slope of the line from the simplified equation. To do this, we can rewrite the equation in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
Subtracting x from both sides of the equation x + y = 6, we get:
y = -x + 6
Now, the equation is in slope-intercept form. We can see that the slope m is -1 and the y-intercept b is 6. The negative slope indicates that the line slopes downward from left to right.
Further Applications of the Simplified Equation
The simplified equation x + y = 6 can be used in various mathematical contexts. Here are a few examples:
- Solving Systems of Equations: If we have another equation involving x and y, we can solve the system of equations using the simplified form. For example, if we have the system:
We can easily solve this system using methods like substitution or elimination.x + y = 6 2x - y = 3 - Graphing: As discussed earlier, the simplified equation makes it easy to graph the line. We can plot the intercepts or use the slope-intercept form to draw the line.
- Optimization Problems: In some optimization problems, we might encounter the equation
x + y = 6as a constraint. The simplified form makes it easier to analyze the feasible region and find optimal solutions.
Importance of Showing Steps
In the simplification process, it's crucial to show each step clearly. This not only helps in avoiding errors but also makes it easier for others to follow the logic and understand the solution. Each step should be justified with the appropriate mathematical rule or operation.
In this article, we have demonstrated a step-by-step approach to simplifying the given equation. By simplifying the fractions and dividing by the common factor, we arrived at the concise form x + y = 6. This simplified form provides a clearer understanding of the relationship between x and y and facilitates further analysis and applications.
Conclusion
Simplifying equations is a fundamental skill in mathematics. It allows us to transform complex expressions into more manageable forms, revealing the underlying structure and relationships. In this article, we successfully simplified the equation:
\frac{4x}{2} + \frac{6y}{3} = 12
to its simplest form:
x + y = 6
By following a step-by-step approach, we demonstrated the techniques involved in simplification, including reducing fractions and dividing by common factors. The simplified equation provides valuable insights into the relationship between x and y and can be used in various mathematical contexts.
In summary, simplification is an essential tool in the mathematician's arsenal, enabling us to solve problems more efficiently and gain a deeper understanding of mathematical concepts. Remember to always show your steps clearly and justify each operation to ensure accuracy and clarity in your solutions. Mastering simplification techniques will undoubtedly enhance your mathematical abilities and problem-solving skills.