There are two numbers such that the sum of thrice the first number and twice the second number
is 180 and the difference of the four times the first number and twice the second number is 100. Then
the product of two numbers is

1. 1000
2. 1200
3. 1800
4. 2000

Introduction to the Problem

Solving mathematical problems involving linear equations is a fundamental concept in algebra. In this article, we will solve a problem involving two unknown numbers by setting up and solving a system of linear equations. This approach is crucial for competitive exams like CSIR NET Life Science, DBT BET JRF, IIT JAM, and GATE Biotechnology.

The problem involves two unknown numbers where:

• The sum of thrice the first number and twice the second number equals 180.
• The difference of four times the first number and twice the second number equals 100.

Let’s solve this step-by-step to find the two numbers and their product.

Key Phrase: Product of Two Numbers Based on Linear Equations

✅ Question and Answer

Question:
There are two numbers such that:

• The sum of thrice the first number and twice the second number is 180.
• The difference of four times the first number and twice the second number is 100.
  What is the product of the two numbers?

Options:

1. 1000
2. 1200
3. 1800
4. 2000

Correct Answer: ✔️ Option 2: 1200

Step-by-Step Solution

Step 1: Define the Variables

Let the two numbers be:

• x = First number
• y = Second number

Step 2: Set Up the Equations Based on the Given Conditions

1. Sum of thrice the first number and twice the second number equals 180:

                      3x+2y=180

2. Difference of four times the first number and twice the second number equals 100:

                    4x−2y=100

Step 3: Solve the System of Equations

Add both equations to eliminate $y$:

(3x+2y)+(4x−2y)=180+100

3x+4x+2y−2y=280

7x=280

x=40

Substituting the value of $x$ into the first equation:

3(40)+2y=180

120+2y=180

2y=180−120=60

y=30

Step 4: Find the Product of Two Numbers

The product of two numbers is:

x×y=40×30=1200

Final Answer:

✔️ The product of the two numbers is 1200.

Why Option 2 (1200) Is Correct

• The system of equations was correctly set up based on the given conditions.
• Solving the system yielded $x=40$ and $y=30$.
• The product of 40 and 30 is 1200, confirming that option 2 is the correct answer.

Why Other Options Are Incorrect

Concepts Involved in Solving the Problem

Linear Equations:

• Linear equations are algebraic expressions where variables have a degree of one.
• A system of linear equations can be solved using substitution, elimination, or matrix methods.

Simultaneous Equations:

• Two linear equations with two variables can be solved simultaneously using elimination or substitution.
• The elimination method is often preferred when coefficients align conveniently.

Algebraic Multiplication:

• Once the values of variables are determined, their product is calculated directly.

Biological Relevance of Linear Equations

1. Enzyme Kinetics:
   • Michaelis-Menten equation is a linear form used to determine enzyme activity.
2. Gene Expression:
   • Linear models are used to analyze gene expression data.
3. Population Growth:
   • Population changes can be modeled using linear equations.

Importance for Competitive Exams

Understanding linear equations is essential for success in:
CSIR NET Life Science
DBT BET JRF
IIT JAM
GATE Biotechnology

Frequently Asked Questions (FAQs)

1. What are linear equations?

Linear equations are algebraic equations where the highest power of the variable is one. They represent straight-line relationships between variables.

2. How can linear equations be solved?

Linear equations can be solved using:

• Substitution
• Elimination
• Matrix methods

3. Why are linear equations important in biology?

Linear models are widely used in biology for modeling enzyme activity, gene expression, and population dynamics.

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Conclusion

By setting up and solving the system of linear equations, we found that the two numbers are 40 and 30. Their product is 1200. Mastering the art of solving linear equations is critical for tackling mathematical problems in competitive exams like CSIR NET, DBT BET JRF, IIT JAM, and GATE Biotechnology.