1. Standard form

A pair of linear equations in x and y can be written as:

A solution is an ordered pair (x,y) satisfying both equations.

2. Graphical meaning

Each equation represents a straight line. Their intersection gives the common solution.

3. Consistency conditions

For a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0:

• Unique solution: a₁/a₂ ≠ b₁/b₂
• No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂

4. Substitution method

Express one variable from one equation and substitute into the other.

5. Elimination method

Multiply one or both equations, if needed, so that one variable has equal coefficients. Add or subtract to eliminate that variable.

6. Cross-multiplication

For a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0:

Arrange the equations correctly before applying the formula; sign errors are common.

7. Real-life application

The sum of two numbers is 52 and their difference is 14.

x+y=52 and x−y=14. Adding gives 2x=66, so x=33 and y=19.

8. Competency Case – School Stationery

A school sells 3 notebooks and 2 pens for ₹86. Another combination of 2 notebooks and 4 pens costs ₹92. Let notebook cost be x and pen cost be y.

1. Form the two equations.
2. Solve using elimination.
3. Interpret the values in the context.

Equations: 3x+2y=86 and 2x+4y=92.

Divide the second equation by 2: x+2y=46, so x=46−2y. Substitute into the first: 3(46−2y)+2y=86 → 138−4y=86 → y=13. Hence x=20.

Notebook = ₹20; Pen = ₹13.

9. Practice Set

1. Solve 2x+3y=11 and 3x−2y=4.
2. Determine whether 3x+2y=5 and 6x+4y=10 have a unique, no, or infinitely many solutions.
3. Find the value of k for which kx+3y=7 and 12x+ky=14 have no solution.
4. The sum of two numbers is 60 and their difference is 18. Find them.
5. A two-digit number has digits whose sum is 9. Reversing the digits increases the number by 27. Find the number.

10. Quick Revision

Pair of lines → intersection → consistency → substitution → elimination → cross-multiplication → applications.