CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables Notes Set 12

Pair of Linear Equations

A pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has a unique solution if a₁/a₂ ≠ b₁/b₂ (intersecting lines), infinitely many solutions if a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines) and no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines).
Methods: substitution, elimination, and cross-multiplication x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁), y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁). Equations such as ax + by = c in the reciprocals 1/x and 1/y are solved by a change of variable.

Word Problems

Choose a variable for each unknown quantity, translate each statement into an equation, solve, and check against the story. Typical types: digits and numbers, fractions, ages, boats in a stream, work, and cost problems.

Solved Examples

Example 1. Solve x + y = 7 and x − y = 3.
Answer:
Adding: 2x = 10, so x = 5 and y = 2.
x = 5, y = 2

Example 2. Solve 2x + 3y = 12 and 3x − y = 7 by substitution.
Answer:
From the second, y = 3x − 7. Then 2x + 9x − 21 = 12, so x = 3 and y = 2.
x = 3, y = 2

Example 3. For what value of k does 3x + 2y = 5 and 6x + ky = 10 have infinitely many solutions?
Answer:
3/6 = 2/k = 5/10, so k = 4.
k = 4

Example 4. For what value of k do kx + 2y = 3 and 4x + 8y = 5 have no solution?
Answer:
k/4 = 2/8 ≠ 3/5, so k = 1.
k = 1

Example 5. The sum of two numbers is 40 and their difference is 12. Find them.
Answer:
x + y = 40 and x − y = 12 give x = 26 and y = 14.
26 and 14

Example 6. The cost of 2 pencils and 3 pens is Rs. 18 and of 3 pencils and 2 pens is Rs. 17. Find the cost of a pen.
Answer:
2p + 3q = 18 and 3p + 2q = 17. Adding: 5(p + q) = 35, so p + q = 7. Subtracting: q − p = 1.
So q = 4 and p = 3. Rs. 4

Example 7. Find the point where the lines x + y = 6 and x − y = 2 meet.
Answer:
Adding: x = 4, so y = 2.
(4, 2)

Example 8. The sum of the ages of a father and his son is 50 years. Five years ago the father was four times as old as the son. Find their present ages.
Answer:
F + S = 50 and F − 5 = 4(S − 5), so 45 − S = 4S − 20 and S = 13.
Son 13 years, father 37 years

Example 9. Solve 2/x + 3/y = 7 and 1/x + 2/y = 4 (x, y not zero).
Answer:
Let u = 1/x and v = 1/y: 2u + 3v = 7 and u + 2v = 4, so v = 1 and u = 2.
x = 1/2, y = 1

Example 10. Solve 3x + 2y = 11 and 2x + 3y = 9.
Answer:
Adding: x + y = 4; subtracting: x − y = 2.
x = 3, y = 1