Multiply (2) by 3: $ 3a + 3b + 3c = 9 $ - Portal da Acústica
Simplifying the Equation: Understanding $ 3a + 3b + 3c = 9 $ in Detail
Simplifying the Equation: Understanding $ 3a + 3b + 3c = 9 $ in Detail
If you've come across the equation $ 3a + 3b + 3c = 9 $, you're not alone—this type of linear expression appears frequently in algebra, especially when solving for variables in real-world applications like budgeting, statistics, and system modeling. In this article, we’ll explore how to simplify and solve this equation step-by-step, and explain its underlying structure to deepen your understanding of linear relationships.
Breaking Down the Equation: $ 3a + 3b + 3c = 9 $
Understanding the Context
At first glance, the expression $ 3a + 3b + 3c $ may seem complex due to the presence of three variables. However, algebraic simplification makes it much clearer.
Step 1: Factor Out the Common Term
Notice that each term on the left side includes the factor $ 3 $. We can factor it out:
$$
3a + 3b + 3c = 3(a + b + c)
$$
So the original equation becomes:
Key Insights
$$
3(a + b + c) = 9
$$
Step 2: Solve for the Parenthetical Expression
Divide both sides of the equation by 3:
$$
a + b + c = rac{9}{3} = 3
$$
This simplifies the equation to:
$$
a + b + c = 3
$$
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📰 These are distinct points. 📰 Thus, all 30 factor pairs produce valid lattice points. 📰 Hence, the total number of lattice points is $ \boxed{30} $.Final Thoughts
What Does This Mean?
The equation $ a + b + c = 3 $ expresses a constraint among the variables $ a $, $ b $, and $ c $. It tells us that the sum of these three quantities must equal 3. This is commonly seen in problems involving:
- Resource allocation: Distributing 3 units among a, b, and c,
- Mean value calculations: When scaling a single value across multiple groups,
- System modeling: Balancing additive contributions contributing to a total.
Solving for One Variable
While the equation defines a relationship, you can isolate any one variable in terms of the others. For example, solving for $ c $:
$$
c = 3 - a - b
$$
This helps in solving systems of equations or substituting values in larger expressions.
Real-World Example
Imagine you're managing a budget allocating $3a, $3b, and $3c across three departments, and the total must be $9 million. The equation tells you the combined funding per department (scaled by 3) equals $9, so each department contributes equally in this context when $ a = b = c = 1 $.
