\[ 400a + 20b + 2 = 50 \] - Parker Core Knowledge
Understanding the Equation: 400a + 20b + 2 = 50 in Linear Algebra
Understanding the Equation: 400a + 20b + 2 = 50 in Linear Algebra
The equation 400a + 20b + 2 = 50 is a linear equation involving two variables, a and b. While seemingly simple, it serves as a foundational concept in mathematics, computer science, economics, and engineering, particularly in fields like optimization, linear programming, and systems modeling. In this article, we’ll break down the equation step-by-step, explore its solutions, and discuss its practical applications.
Understanding the Context
What Is the Equation: 400a + 20b + 2 = 50?
The expression
400a + 20b + 2 = 50
is a linear equation in two variables. Rearranging it into standard form:
400a + 20b = 48
This equation describes a straight line in a 2D coordinate system where a is the independent variable and b is the dependent variable. However, since both variables appear linearly with non-zero coefficients, this equation lies on a linear function rather than a single y-value for a fixed x—indicating relationships between two evolving quantities a and b.
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Key Insights
Step-by-Step Solution
Step 1: Simplify the equation
Start by isolating the variable terms:
400a + 20b = 48
Factor out the greatest common divisor (GCD) of the coefficients. The GCD of 400 and 20 is 20:
20(20a + b) = 48
Divide both sides by 20:
20a + b = 2.4
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This simplification helps interpret the relationship between a and b.
Step 2: Express b in terms of a (or vice versa)
From 20a + b = 2.4, solve for b:
b = 2.4 - 20a
Similarly, solve for a:
a = (2.4 - b)/20 = 0.12 - 0.05b
This linear relationship allows substitution into other equations, modeling dependencies in real-world systems.
Step 3: Identify Solutions and Constraints
The equation defines an infinite set of (a, b) pairs satisfying 20a + b = 2.4. It is underdetermined (two variables, one equation), so solutions are parametric.
For example:
- If a = 0 → b = 2.4
- If a = 0.01 → b = 2.4 - 20(0.01) = 2.0
- If b = 0 → a = 0.12
This line represents trade-offs: increasing a decreases b by a scaled amount (20:1), and vice versa.
