1. Chapter at a Glance
"Two pens and three pencils cost ₹40." How much is one pen? We cannot tell — there are many possible answers: (₹14, ₹4), (₹11, ₹6), (₹5, ₹10)… Each answer is an ordered pair, and when we plot all of them, they lie on one straight line. That is why an equation like 2x + 3y = 40 is called a linear equation in two variables. If we get a second clue, the two lines usually cross at one point — the unique answer. This chapter builds these ideas: solutions, graphs, slope, and solving pairs of linear equations by substitution, elimination and graphs.
- Linear equations ax + by + c = 0 and their solutions as ordered pairs.
- Why there are infinitely many solutions — and why they lie on a line.
- Slope and the slope–intercept form y = mx + c.
- Special lines: x = a, y = b, lines through the origin.
- Pairs of linear equations: graphical, substitution and elimination methods.
- Unique, no, or infinitely many solutions — algebraic conditions and geometric meaning.
2. Linear Equations in Two Variables
An equation of the form ax + by + c = 0, where a, b, c are real numbers and a, b are not both zero, is a linear equation in two variables.
A solution is a pair (x, y) that makes the equation true. To find solutions, pick any value of x and solve for y.
Find four solutions of 2x + 3y = 12.
Solution: x = 0 → y = 4; x = 3 → y = 2; x = 6 → y = 0; x = −3 → y = 6. Solutions: (0, 4), (3, 2), (6, 0), (−3, 6).
Since x can be any real number, the equation has infinitely many solutions.
3. The Graph Is a Straight Line
Plot the solutions and they lie on a straight line. Every point on the line is a solution, and every solution is a point on the line.
- x-intercept: put y = 0. y-intercept: put x = 0. These two points are often the quickest way to draw the line.
- x = a is a vertical line; y = b is a horizontal line. x = 0 is the y-axis, y = 0 is the x-axis.
- y = mx passes through the origin.
4. Slope and the Form y = mx + c
The slope (gradient) of a line measures its steepness: how much y changes when x increases by 1.
Through (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁) = rise ÷ run.
Slope–intercept form: y = mx + c, where m is the slope and c is the y-intercept.
| Slope | Line |
|---|---|
| m > 0 | Rises from left to right |
| m < 0 | Falls from left to right |
| m = 0 | Horizontal |
| Undefined | Vertical (x = a) |
Writing ax + by + c = 0 as y = (−a/b)x − c/b gives slope −a/b. Lines with the same slope are parallel.
Find the slope and y-intercept of 4x − 2y + 6 = 0.
Solution: 2y = 4x + 6 → y = 2x + 3. Slope 2, y-intercept 3.
5. Pairs of Linear Equations
Two linear equations in the same two variables form a pair (or system). A solution of the pair satisfies both equations — it is a point on both lines.
Graphical method
Draw both lines; read off the intersection point. Good for seeing what is happening; less accurate when the answer is not a whole number.
Substitution method
Solve x − y = 1 and 2x + 3y = 12.
Solution: From the first, x = y + 1. Substitute: 2(y + 1) + 3y = 12 → 5y = 10 → y = 2, x = 3. Solution (3, 2) — the point where the lines in the graph above cross.
Elimination method
Solve 3x + 4y = 10 and 2x − 2y = 2.
Solution: Multiply the second by 2: 4x − 4y = 4. Add to the first: 7x = 14 → x = 2. Then 2(2) − 2y = 2 → y = 1. Solution (2, 1).
6. How Many Solutions?
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
| Condition | Lines | Solutions | Called |
|---|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting | Exactly one | Consistent |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel | None | Inconsistent |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident (same line) | Infinitely many | Dependent (consistent) |
Classify: (a) 2x + 3y = 7, 4x + 6y = 9 (b) x + 2y = 4, 3x + 6y = 12.
Solution: (a) 2/4 = 3/6 = ½ but 7/9 ≠ ½ → parallel, no solution. (b) 1/3 = 2/6 = 4/12 → same line, infinitely many.
7. Word Problems
Adult tickets cost ₹50 and child tickets ₹30. 120 tickets earned ₹4,800. How many of each?
Solution: a + c = 120, 50a + 30c = 4800. Substitute c = 120 − a: 50a + 3600 − 30a = 4800 → a = 60, c = 60.
Puzzles of this type appear in old Indian arithmetic texts: "The price of 9 citrons and 7 wood-apples is 107; of 7 citrons and 9 wood-apples is 101. Find the price of each."
Solution: 9c + 7w = 107, 7c + 9w = 101. Add: 16(c + w) = 208 → c + w = 13. Subtract: 2(c − w) = 6 → c − w = 3. So c = 8, w = 5.
8. Solved Problems (Board Pattern)
For what k does kx + 3y = 5, 6x + 9y = 11 have no solution?
Solution: k/6 = 3/9 → k = 2; check 5/11 ≠ 1/3 ✓. k = 2.
Five years ago, A was three times as old as B. Ten years later, A will be twice as old as B. Find their present ages.
Solution: A − 5 = 3(B − 5) → A = 3B − 10. A + 10 = 2(B + 10) → A = 2B + 10. So 3B − 10 = 2B + 10 → B = 20, A = 50.
Find the equation of the line through (1, 5) and (3, 9).
Solution: m = 4/2 = 2; 5 = 2(1) + c → c = 3. y = 2x + 3.
9. Master Formula Table
| Idea | Formula |
|---|---|
| General linear equation | ax + by + c = 0 |
| Slope through two points | m = (y₂ − y₁)/(x₂ − x₁) |
| Slope–intercept form | y = mx + c |
| Slope of ax + by + c = 0 | −a/b |
| Intercepts | x-int: −c/a, y-int: −c/b |
| Unique solution | a₁/a₂ ≠ b₁/b₂ |
| No solution | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ |
| Infinitely many | a₁/a₂ = b₁/b₂ = c₁/c₂ |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| One equation | Infinitely many solutions on a line | Find solutions; check a point |
| Graph | Use intercepts | Draw and read lines |
| Slope | Rise ÷ run | Parallel lines, y = mx + c |
| Substitution | Express one variable, replace | Simple coefficients |
| Elimination | Make coefficients equal, add/subtract | Larger coefficients |
| Consistency | Compare ratios | Find k for 0/1/∞ solutions |
11. Mind Map
- ax + by + c = 0
- ∞ solutions
- Intercepts
- Slope, y = mx + c
- Graph
- Substitution, elimination
- Intersect: 1
- Parallel 0, same ∞
12. Tips and Tricks
- If the coefficients are swapped (9c + 7w, 7c + 9w), add and subtract the equations.
- Always check your solution in both original equations.
- When drawing by hand, choose x-values that make y a whole number.
- Forgetting to multiply every term when scaling an equation for elimination.
- Confusing "no solution" (parallel) with "infinitely many" (same line).
- Getting the slope sign wrong: slope of ax + by + c = 0 is −a/b.
13. Important One-liners
- A linear equation in two variables has infinitely many solutions.
- Its graph is a straight line.
- Parallel lines have equal slopes.
- Two intersecting lines give exactly one solution.
- The graph of x = 3 is a vertical line.
- Every point on the graph is a solution, and vice versa.
14. Exam-pattern Questions with Detailed Answers
Is (2, −1) a solution of 3x + 4y = 2?
Show answer
6 − 4 = 2 ✓.
Answer: Yes
How many solutions does x + y = 5 have?
- One
- Two
- None
- Infinitely many
Show answer
A single linear equation in two variables has infinitely many solutions.
Answer: (D)
Find the slope of the line through (2, 3) and (6, 11).
Show answer
(11 − 3)/(6 − 2) = 2.
Answer: 2
Solve by elimination: 5x + 2y = 16, 3x − 2y = 0.
Show answer
Add: 8x = 16 → x = 2; 3(2) = 2y → y = 3.
Answer: (2, 3)
Find k if (3, k) lies on 2x − 5y = 1.
Show answer
6 − 5k = 1 → k = 1.
Answer: k = 1
Without solving, state the number of solutions of 3x − y = 4 and 6x − 2y = 8.
Show answer
3/6 = −1/−2 = 4/8 = ½ → coincident lines, infinitely many solutions.
Answer: Infinitely many
The sum of two numbers is 50 and their difference is 14. Find them.
Show answer
x + y = 50, x − y = 14 → 2x = 64 → x = 32, y = 18.
Answer: 32 and 18
Write the equation of the line parallel to y = 3x − 2 passing through (0, 5).
Show answer
Same slope 3, y-intercept 5: y = 3x + 5.
Answer: y = 3x + 5
For what values of a and b does 2x + 3y = 7 and (a − b)x + (a + b)y = 3a + b − 2 have infinitely many solutions?
Show answer
2/(a − b) = 3/(a + b) = 7/(3a + b − 2). From the first two: 2a + 2b = 3a − 3b → a = 5b. From 2/(a − b) = 7/(3a + b − 2): 6a + 2b − 4 = 7a − 7b → a = 9b − 4. So 5b = 9b − 4 → b = 1, a = 5.
Answer: a = 5, b = 1
A fraction becomes 9/11 if 2 is added to both numerator and denominator. If 3 is added to both, it becomes 5/6. Find the fraction.
Show answer
11(x + 2) = 9(y + 2) → 11x − 9y = −4. 6(x + 3) = 5(y + 3) → 6x − 5y = −3. Multiply: 55x − 45y = −20 and 54x − 45y = −27 → x = 7, y = 9.
Answer: 7/9
A boat goes 30 km upstream and 44 km downstream in 10 hours. It goes 40 km upstream and 55 km downstream in 13 hours. Find the speed of the boat in still water and of the stream.
Show answer
Let u = 1/(upstream speed), v = 1/(downstream speed). 30u + 44v = 10, 40u + 55v = 13. Multiply first by 4, second by 3: 120u + 176v = 40, 120u + 165v = 39 → 11v = 1 → v = 1/11, so downstream = 11 km/h. Then 30u = 10 − 4 = 6 → u = 1/5, upstream = 5 km/h. Boat = 8 km/h, stream = 3 km/h.
Answer: Boat 8 km/h, stream 3 km/h
Find the area of the triangle formed by the lines x + y = 6, x − y = 2 and the x-axis.
Show answer
Intersection: x = 4, y = 2. x-intercepts: x + y = 6 → (6, 0); x − y = 2 → (2, 0). Base = 4, height = 2. Area = ½ × 4 × 2 = 4 square units.
Answer: 4 square units
15. Quick Revision Summary
- ax + by + c = 0 has infinitely many solutions, all on one straight line.
- Slope m = rise/run; y = mx + c; parallel lines share slopes.
- Pairs: solve by graph, substitution or elimination; check in both equations.
- Compare a₁/a₂, b₁/b₂, c₁/c₂ to decide 1, 0 or ∞ solutions.
- Translate word problems into two equations in two unknowns.