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.

\[ 400a + 20b + 2 = 50 \]

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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\min (2\sin x + 3\cos x + 4) = 4 + \min(2\sin x + 3\cos x) = 4 + (-\sqrt{13}) = 4 - \sqrt{13}
And since this value is positive, the absolute value does not introduce a smaller value.
Thus, the minimum value is $ \boxed{4 - \sqrt{13}} $.

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Final Thoughts

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.