10t + 25b = 155
Understanding the Equation 10t + 25b = 155: A Complete Breakdown
Welcome to our in-depth SEO article exploring the equation 10t + 25b = 155. Whether you're a student tackling math problems, a teacher explaining linear equations, or a curious learner curious about financial or real-world applications of algebra, this guide will break down the equation step-by-step while optimizing for search engines.
What Is the Equation 10t + 25b = 155?
The equation 10t + 25b = 155 is a linear Diophantine equation involving two variables:
- t and b represent unknown quantities.
- The coefficients 10 and 25 multiply these variables.
- The constant 155 is the total value on the right-hand side.
This type of equation appears in various fields such as economics, budgeting, scalability modeling, and even in physics when balancing weighted averages or ratios.
Why Does This Equation Matter?
Integer solutions to linear equations like 10t + 25b = 155 help in:
- Optimizing resource allocation where two variables cost different amounts.
- Modeling real-life scenarios like splitting payments between two parties (e.g., splitting a bill or funding).
- Teaching foundational algebra skills and Diophantine problem-solving.
Step-by-Step Solution to Solve 10t + 25b = 155
Step 1: Simplify the Equation
Check if the equation is reducible by finding the greatest common divisor (GCD) of 10 and 25.
- GCD(10, 25) = 5, which divides 155 exactly: 155 ÷ 5 = 31. This confirms integer solutions exist.
Divide the entire equation by 5: 10t + 25b = 155 → 2t + 5b = 31
Step 2: Solve for One Variable
Solve for t in terms of b: 2t = 31 - 5b t = (31 - 5b) / 2
Step 3: Find Integer Solutions
Since t and b must be integers (e.g., counting items), 31 - 5b must be even. Try integer values of b such that (31 - 5b) is divisible by 2:
- b = 1 → t = (31 - 5×1) / 2 = 26/2 = 13 ✔
- b = 3 → t = (31 - 15)/2 = 16/2 = 8 ✔
- b = 5 → t = (31 - 25)/2 = 6/2 = 3 ✔
- b = 7 → t = (31 - 35)/2 = negative → invalid
Valid integer solutions include:
- (t, b) = (13, 1)
- (8, 3)
- (3, 5)
Real-World Applications of 10t + 25b = 155
1. Splitting a Bill Between Two People
Imagine person A pays 10 units per item, person B pays 25 units per item, and together they spend exactly 155 units. The equation models how many items each buys under fixed-price contributions.
2. Budgeting for Supplies
Suppose a small business buys two types of supplies: one costing 10 per unit, another 25 per unit, totaling 155. This equation helps plan purchases within a fixed budget.
3. Profit Sharing Between Investors
Partners in a project contribute capital such that combined contributions matching a scaled profit share equal 155. Solving confirms balanced investment ratios.
Math Tip: Graphing Integer Solutions
Plotting the simplified equation 2t + 5b = 31 reveals lattice points—a visual way to identify all integer pairs (t, b) that satisfy the equation. These visualizations are valuable for teaching number theory and discrete math.
How to Use This Equation in Problem Solving
- Identify constraints on t and b (e.g., non-negative integers).
- Simplify using GCD to reduce complexity.
- Parameterize one variable to express all solutions.
- Plug in values to find feasible pairs within real-world bounds.
- Verify using substitution.
Conclusion
The equation 10t + 25b = 155 is a typical linear Diophantine equation that demonstrates how two variables with different weights sum to a fixed total. Solving it develops algebraic reasoning and problem-solving skills applicable across sciences, economics, and daily life. Whether splitting costs, allocating budgets, or modeling investments, understanding such equations unlocks smarter decision-making.
Frequently Asked Questions (FAQs)
Q: Why does GCD matter in solving 10t + 25b = 155? A: Dividing the equation by the GCD (5) ensures integer solutions are possible and simplifies computation.
Q: Can t and b be non-integers? A: The equation inherently supports integer solutions in practical contexts; fractional values may lack meaning in real-world scenarios.
Q: How can I graph all integer solutions? A: Solve for t in terms of b, then substitute integer values of b such that t remains an integer, plotting the resulting lattice points.
Q: What’s a practical use of this equation outside math? A: It models fair cost splitting, budget allocation, and scalable project funding in business and personal finance.
SEO Keyword Strategy
- Primary keywords: linear equation 10t + 25b = 155, solving integer solutions, Diophantine equation,
- Related keywords: fields of application linear equations, budgeting algebra, real-life equation problems, 2t + 5b = 31,
- Included long-tail phrases like how to solve 10t + 25b = 155 with integer solutions.
Keywords optimized for search: 10t + 25b = 155 equation solution, Diophantine equation example, solve 10t + 25b = 155, integer solutions 2t + 5b = 31, real-world applications of 10t + 25b
Stay curious and keep solving — every equation tells a story!
