Understanding the Equation: k = 5(7b + 3) + 3 = 35b + 18

When faced with a mathematical expression like k = 5(7b + 3) + 3, solving and simplifying it can feel challenging at first—but with a clear step-by-step approach, the process becomes manageable and even insightful. In this article, we’ll explore how to simplify the equation k = 5(7b + 3) + 3, break it down to k = 35b + 18, and offer practical tips for solving linear equations involving variables like b.

Understanding the Context

Step-by-Step Breakdown of the Equation

Start with the original expression:
k = 5(7b + 3) + 3

  1. Apply the distributive property:
    Multiply 5 across the terms inside the parentheses:
    k = 5 × 7b + 5 × 3 + 3
    k = 35b + 15 + 3

  2. Combine like terms:
    Add 15 and 3:
    k = 35b + 18

k = 5(7b + 3) + 3 = 35b + 15 + 3 = 35b + 18

Key Insights

This simplification reveals a linear equation in standard form:
k = 35b + 18

Why Simplify Linear Equations Like This?

Simplifying equations into forms such as k = 35b + 18 is essential in algebra for several reasons:

  • Easier graphing: The equation represents a straight line on a coordinate plane, where the slope (35) and y-intercept (18) become immediately clear.
  • Efficient solving: Once simplified, substituting values for b becomes straightforward, helping solve for unknowns quickly.
  • Foundation for advanced math: Linear equations form the basis for systems of equations, calculus, and algebraic modeling in real-world applications.

Continue Reading

So, the base-ten equivalent is:
oxed{1244}
Question:** Find the smallest positive integer \(n\) such that \(n^3\) ends in the digits \(888\).

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Final Thoughts

How to Use This Equation in Real-World Contexts

Equations like k = 35b + 18 are not just abstract—they model real-life scenarios. For example:

  • Business: If k represents total cost and b is the number of units produced, the equation shows fixed costs (18) plus a variable cost scaling with b at a rate of 35 per unit.
  • Physics: Think of k as total distance traveled, b as time, and the equation capturing motion with constant speed plus initial offset.

Solving for b: Practical Applications

To isolate b, start from the simplified form:
k = 35b + 18

Subtract 18:
k – 18 = 35b

Then divide by 35:
b = (k – 18) / 35

This form helps answer “What value of b produces a given k—an essential skill for data analysis, forecasting, and algebra-based problem solving.