Linear Equations in Two Variables Notes for SSC Exams

A linear equation in two variables has the form ax + by + c = 0, and two such equations together are called simultaneous equations. SSC exams use them in direct questions and in word problems on numbers, ages, digits, fractions and speed. These linear equations notes for SSC explain the three methods, the conditions for solutions and the word problem types with solved examples and practice questions.

Standard Form

A pair of linear equations is written as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. The graph of each equation is a straight line.

Types of Solutions

Methods of Solving

Solved Examples

Example 1 (substitution). Solve 3x + 2y = 16 and x + y = 7.
From the second equation x = 7 − y. Then 3 (7 − y) + 2y = 16 gives 21 − y = 16, so y = 5 and x = 2. Answer: x = 2, y = 5.

Example 2 (elimination). Solve x + y = 10 and x − y = 2.
Adding the equations gives 2x = 12, so x = 6, and then y = 4.

Example 3 (condition). For which value of k does 2x + 3y = 5 and 4x + 6y = k have infinitely many solutions?
The first two ratios are 2/4 = 3/6 = 1/2, so 5/k must also be 1/2, which gives k = 10.

Word Problems

Numbers. The sum of two numbers is 40 and their difference is 12. The numbers are (40 + 12) / 2 = 26 and (40 − 12) / 2 = 14.

Ages. A father is 3 times as old as his son. After 5 years, the sum of their ages is 70. Let the son be s years, so the father is 3s. Then (3s + 5) + (s + 5) = 70, which gives s = 15. The son is 15 and the father is 45.

Digits. The sum of the digits of a two-digit number is 9, and the number obtained by reversing the digits is 27 more than the original number. Let the digits be x and y. Then x + y = 9 and 9 (y − x) = 27, so y − x = 3. This gives y = 6 and x = 3, and the number is 36.

Common Word Problem Types

Quick Tips

• Choose the method that needs the least work: elimination when the coefficients are easy, substitution when one variable has the coefficient 1.
• Check your answer by putting the values in both equations.
• Write what x and y stand for before you start a word problem.
• Use the condition table for questions about the number of solutions.

Quick Revision Points

• The standard form is a₁x + b₁y + c₁ = 0.
• Unique solution when a₁/a₂ ≠ b₁/b₂.
• No solution for parallel lines.
• Infinite solutions for coincident lines.
• Digit problem: number = 10x + y.

Practice Questions With Answers

1. Solve x + y = 12 and x − y = 4.
   Answer: x = 8, y = 4
2. Solve 2x + y = 11 and x + y = 7.
   Answer: x = 4, y = 3
3. The sum of two numbers is 50 and the difference is 10. Find the numbers.
   Answer: 30 and 20
4. For which value of k does 3x + 2y = 7 and 6x + 4y = k have infinitely many solutions?
   Answer: 14
5. Do the lines x + y = 5 and 2x + 2y = 7 meet?
   Answer: No (parallel lines)
6. The sum of the digits of a two-digit number is 7 and the tens digit is 3 more than the units digit. Find the number.
   Answer: 52
7. The cost of 2 pens and 3 books is Rs 50, and the cost of 1 pen and 1 book is Rs 20. Find the cost of one pen.
   Answer: Rs 10
8. Two numbers are in the ratio 3 : 2 and their sum is 100. Find them.
   Answer: 60 and 40
9. The ages of A and B are in the ratio 5 : 3 and the sum is 64. Find A's age.
   Answer: 40 years
10. How many solutions does x + y = 4 and 2x + 2y = 8 have?
    Answer: Infinitely many

How to Practise This Chapter

Solve five equation problems daily: two direct and three word problems. Then attempt timed sets from our practice papers and the previous year papers page. Read the previous chapter, Arithmetic and Geometric Progression, and the next chapter, Algebraic Identities and Factorisation. See all topics on the Mathematics notes page, and also read Problems on Ages.

Frequently Asked Questions

What is a linear equation in two variables?

It is an equation of the form ax plus by plus c equals 0, whose graph is a straight line.

What are simultaneous equations?

Simultaneous equations are two or more equations that are solved together to find the common values of the variables.

What are the methods to solve linear equations?

The methods are substitution, elimination, cross multiplication and the graphical method.

When do two linear equations have a unique solution?

They have a unique solution when the ratio of the x coefficients is not equal to the ratio of the y coefficients.

When do two linear equations have no solution?

They have no solution when the lines are parallel, that is, when the ratio of the coefficients is equal but the constants are in a different ratio.

When do two linear equations have infinite solutions?

They have infinitely many solutions when all three ratios are equal, so that the two lines are the same line.

Which method is the fastest?

Elimination is usually the fastest when the coefficients are easy, and substitution is fast when one variable has the coefficient 1.

How do I write a digit problem as equations?

Take the digits as x and y, write the number as 10x plus y and the reversed number as 10y plus x.

How can I check my answer?

Put the values of x and y in both the given equations and check that both are true.

Are word problems on equations asked in SSC?

Yes. Problems on numbers, ages, digits and costs are asked in the SSC maths section.

What is the graph of a linear equation?

The graph of a linear equation in two variables is a straight line.

How many questions come from linear equations in SSC exams?

Usually two or three questions come from linear equations and word problems, but the number changes from year to year.