What is a Word Problem on Linear Equations?

S6-SA1-0132

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A word problem on linear equations is a real-life situation described in words, which you need to translate into mathematical equations. These equations will involve one or more variables, and their highest power will be one, allowing you to solve for the unknown values.

Simple Example

Quick Example

Imagine you bought 2 packets of biscuits and 1 packet of namkeen for a total of Rs 70. If the namkeen packet cost Rs 30, how much did one packet of biscuits cost? Here, the cost of biscuits is the unknown you need to find.

Worked Example

Step-by-Step

QUESTION: A mobile data plan charges a fixed monthly fee of Rs 200 plus Rs 10 for every extra GB of data used. If your bill for a month was Rs 350, how many extra GBs of data did you use?

---1. Understand the unknowns: Let 'x' be the number of extra GBs of data used.

---2. Identify the fixed and variable costs: Fixed monthly fee = Rs 200. Variable cost = Rs 10 per extra GB.

---3. Formulate the equation: Total Bill = Fixed Fee + (Cost per GB * Number of extra GBs)
350 = 200 + (10 * x)

---4. Simplify the equation: 350 = 200 + 10x

---5. Isolate the variable term: Subtract 200 from both sides.
350 - 200 = 10x
150 = 10x

---6. Solve for x: Divide both sides by 10.
x = 150 / 10
x = 15

---ANSWER: You used 15 extra GBs of data.

Why It Matters

Understanding linear equations helps engineers design structures, scientists model physical phenomena, and even AI/ML specialists develop algorithms that predict outcomes. From optimizing delivery routes to managing resources in biotechnology, linear equations are fundamental for solving complex real-world problems and are crucial for careers in fields like data science, engineering, and finance.

Common Mistakes

MISTAKE: Not correctly identifying the variables and constants from the word problem. | CORRECTION: Read the problem carefully, underline the unknown quantities (variables) and the known fixed values (constants). Assign letters like 'x' or 'y' to the variables.

MISTAKE: Setting up the equation incorrectly, often mixing up addition/subtraction or multiplication/division. | CORRECTION: Break down the problem into smaller parts. Think about what operations connect the different pieces of information. For example, 'total' often means addition, 'per' often means multiplication.

MISTAKE: Forgetting to check if the answer makes sense in the context of the original word problem. | CORRECTION: Once you get an answer, plug it back into the original problem's scenario. Does a negative number of students make sense? Does a very high price for a small item seem right? This helps catch calculation errors.

Practice Questions

Try It Yourself

QUESTION: The sum of two numbers is 60. If one number is twice the other, find the two numbers. | ANSWER: The numbers are 20 and 40.

QUESTION: A vendor sells samosas for Rs 10 each and jalebis for Rs 50 per plate. If he sold 15 items in total and earned Rs 350, how many samosas did he sell? | ANSWER: He sold 10 samosas.

QUESTION: An auto-rickshaw charges a base fare of Rs 25 for the first 2 km and Rs 12 for every subsequent kilometer. If a passenger paid Rs 109 for a ride, what was the total distance covered? | ANSWER: The total distance covered was 9 km.

MCQ

Quick Quiz

Which of the following scenarios can be best represented by a linear equation?

Calculating the area of a circle given its radius.

Finding the speed of a car if it travels a fixed distance in varying times.

Determining the total cost of mobile recharge with a fixed base price and a per-minute charge.

Modeling the growth of bacteria that doubles every hour.

The Correct Answer Is:

Option C describes a fixed initial value (base price) and a variable cost that changes linearly with the number of minutes, which is a classic linear equation setup. Options A, B, and D involve quadratic, inverse, or exponential relationships respectively.

Real World Connection

In the Real World

Linear equations are used by delivery services like Zomato or Swiggy to calculate the total delivery cost based on a base fee and distance. Traffic engineers use them to model traffic flow and optimize signal timings. Even ISRO scientists use principles of linear algebra to calculate satellite trajectories and manage fuel consumption.

Key Vocabulary

Key Terms

VARIABLE: An unknown quantity, usually represented by a letter (like x or y), whose value can change. | CONSTANT: A fixed value that does not change. | COEFFICIENT: A number that multiplies a variable (e.g., 5 in 5x). | EQUATION: A mathematical statement showing that two expressions are equal. | SOLUTION: The value(s) of the variable(s) that make the equation true.

What's Next

What to Learn Next

Great job understanding word problems on linear equations! Next, you can explore 'Systems of Linear Equations in Two Variables'. This will teach you how to solve problems with two unknowns and two equations, which is super useful for more complex real-world scenarios.