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medium difficulty8 min

Reducing word problems to linear equations

Reducing a word problem means converting the given statements into two linear equations in two variables.

In one line: Words first, equations next, answer last.

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Understand it

In such problems, the words describe two unknown quantities and two conditions. The first task is to choose variables for the unknowns and turn the statements into equations. After that, the pair can be solved by any suitable algebraic method.

Example

If the sum of two numbers is 20 and their difference is 4, we can write x + y = 20 and x - y = 4.

Formula and rule

Use two unknowns for two conditions, and translate keywords like sum, difference, total, more than, and less than into equations carefully.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

For 'The sum of two numbers is 20 and their difference is 4', let x and y be the numbers.

  1. 1Then x + y = 20 and x - y = 4.
  2. 2Solving gives x = 12 and y = 8.

Method you can reuse

  1. 1Read the problem carefully.
  2. 2Choose variables for the unknowns.
  3. 3Convert each statement into an equation.
  4. 4Solve the pair.
  5. 5Check whether the final answer fits the story.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the first useful step when converting a word problem into a pair of linear equations?

Think first, then show answer
The first useful step is to choose clear variables for the unknown quantities. After that, each statement is translated into one equation based on the given information.

Common wrong answer

A common wrong answer is to use one variable for both quantities. That makes the two unknowns indistinguishable and the equations impossible to interpret.

Don't mix this up

Students sometimes choose the wrong variables or translate the same statement twice instead of forming two different equations.

Exam tip

Read the problem twice: once to choose variables and once to catch the two separate conditions.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

The exam often gives age, number, money, or speed problems and asks students to form equations before solving them.
  1. 1

    Enough for 1 mark

    Reducing a word problem means converting the given statements into two linear equations in two variables.

  2. 2

    Add for 2 marks

    Example: If the sum of two numbers is 20 and their difference is 4, we can write x + y = 20 and x - y = 4.

  3. 3

    Add for 3 marks

    In such problems, the words describe two unknown quantities and two conditions.

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