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Unit III · Pair of linear equations in two variables
6 min readPathway sheet2-mark or 3-mark question

Solving by substitution

Make one variable the subject of one equation, put it into the other, and solve. Then substitute back to find the second variable.

Step 1From the simpler equation, write y in terms of x (or x in terms of y)
Step 2Put it into the other equation, which now has one variable
Best whenA variable has coefficient 1 or −1
Special resultsA false line like 6 = 8 means no solution; a true line like 0 = 0 means infinitely many

Method

  1. 1
    Pick the equation where one variable is easy to separate (coefficient 1 or −1).
  2. 2
    Write that variable in terms of the other, e.g. y = 7 − 2x.
  3. 3★
    Substitute this into the other equation, using brackets. Now it has only one variable.
  4. 4
    Solve for that variable.
  5. 5
    Substitute the value back to find the other variable.
  6. 6
    Check both values in both original equations.

★ key step

Example 1: a direct pair

Problem

Solve by substitution: x − 3y = 1 and 2x + y = 9.

Working
  1. From the first equation: x = 1 + 3y.
  2. Substitute in the second: 2(1 + 3y) + y = 9
  3. 2 + 6y + y = 9, so 7y = 7 and y = 1.
  4. Substitute back: x = 1 + 3(1) = 4.
  5. Check: 4 − 3(1) = 1 and 2(4) + 1 = 9.
Answer

x = 4, y = 1

Example 2: a word problem on ages

Problem

A father is 3 times as old as his son. After 12 years, he will be twice as old as his son. Find their present ages.

Working
  1. Let the son's present age be x years and the father's be y years.
  2. First condition: y = 3x.
  3. After 12 years: y + 12 = 2(x + 12).
  4. Substitute y = 3x: 3x + 12 = 2x + 24.
  5. So x = 12, and y = 3 × 12 = 36.
  6. Check: after 12 years they are 24 and 48, and 48 = 2 × 24.
Answer

The son is 12 years old and the father is 36 years old.

Example 3: when substitution gives a false statement

Problem

Solve by substitution: x + 2y = 3 and 2x + 4y = 8.

Working
  1. From the first equation: x = 3 − 2y.
  2. Substitute in the second: 2(3 − 2y) + 4y = 8
  3. 6 − 4y + 4y = 8, so 6 = 8.
  4. This is false for every value of y.
Answer

The pair has no solution (the lines are parallel).

Avoid these mistakes

  • Use brackets when you substitute: 2(3 − 2y), not 2 × 3 − 2y.
  • Substitute into the other equation, not the one you just used.
  • Find both x and y. Half an answer gets half the marks.
  • Check in both original equations before you box the answer.
In real life

At a mela, suppose a ride ticket costs ₹ 10 more than a snack coupon. Writing ticket = coupon + 10 and putting it into the total bill is the substitution method.

Exam angle

1 marks: “If x = a, y = b is the solution of x + y = 5 and x − y = 1, find a.” From x − y = 1, x = 1 + y. Then 1 + 2y = 5, so y = 2 and x = 3. So a = 3.

2 marks: “Solve by substitution: x + y = 16 and x − y = 6.” x = 16 − y. Then 16 − 2y = 6, so y = 5 and x = 11.

3 marks: “Solve by substitution: 3x + 2y = 12 and x − y = −1. Hence find p if y = px + 1.” x = y − 1. Then 3(y − 1) + 2y = 12, so y = 3 and x = 2. Then 3 = 2p + 1, so p = 1.

Remember it

Isolate, insert, solve, return, check.

Read alongside the books on your syllabus: NCERT, Mathematics: Textbook for Class X · NCERT, Exemplar Problems: Mathematics, Class X. Written from standard textbooks; faculty review in progress. Spotted an error? Report it.