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CBSE · Class 10 · Mathematics

Pair of Linear Equations 3.2: Graphical Method of Solution

5 concepts · 40 practice questions with answers · Free to read

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Type of lines from the ratios

From daily life

Two roads either cross once, never meet, or are really the same road. A pair of linear equations behaves like two such lines.

Simple definition

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: if a₁/a₂ ≠ b₁/b₂ the lines intersect; if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ they are parallel; if a₁/a₂ = b₁/b₂ = c₁/c₂ they coincide.

Formula: a₁/a₂ vs b₁/b₂ vs c₁/c₂

intersectparallelcoincident1 solutionno solutioninfinite

In the figure: Intersecting lines meet once, parallel lines never, coincident lines everywhere.

Rule 1: Compare the ratios — Write the ratios of a, b and c in the same order.

  • Easy: x + y = 5, x − y = 1: 1/1 ≠ 1/(−1), intersect.
  • Medium: 2x + 3y = 6, 4x + 6y = 7: 2/4 = 3/6 ≠ 6/7, parallel.
  • Exam-type: x + 2y = 4, 2x + 4y = 8: all ratios 1/2, coincident.

Rule 2: Number of solutions — Intersect: one. Parallel: none. Coincident: infinitely many.

  • Easy: Intersecting pair: exactly one solution.
  • Medium: Parallel pair: no solution.
  • Exam-type: Coincident pair: infinitely many solutions.
Remember

Always bring both equations to the form ax + by + c = 0 before taking the ratios.

Common mistake

Taking c₁/c₂ with the constant on the right side without changing its sign.

Memory clue

"Different ratios cross, same a-b ratios and different c are parallel."

Practice questions

Easy
  1. 1Which condition is satisfied by the pair 7x − 4y − 4 = 0 and 14x − 8y − 8 = 0?
  2. 2The pair a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 has infinitely many solutions when ___.
  3. 3Which condition is satisfied by the pair −5x − 4y − 3 = 0 and 10x + 8y + 7 = 0?
Medium
  1. 4Which condition is satisfied by the pair 6x + 5y − 3 = 0 and −x + 6y + 1 = 0?
  2. 5Which condition is satisfied by the pair 6x + 2y + 1 = 0 and 6x − 6y − 7 = 0?
  3. 6Which condition is satisfied by the pair −2x + 5y + 6 = 0 and −4x + 10y + 12 = 0?
Challenging
  1. 7Which condition is satisfied by the pair −5x + 6y + 5 = 0 and −15x + 18y + 15 = 0?
  2. 8Which condition is satisfied by the pair −2x − y − 4 = 0 and −4x − 2y − 8 = 0?
Show answers (try them yourself first)
  1. 1a1/a2 = b1/b2 = c1/c2
  2. 2a1/a2 = b1/b2 = c1/c2
  3. 3a1/a2 = b1/b2 ≠ c1/c2
  4. 4a1/a2 ≠ b1/b2
  5. 5a1/a2 ≠ b1/b2
  6. 6a1/a2 = b1/b2 = c1/c2
  7. 7a1/a2 = b1/b2 = c1/c2
  8. 8a1/a2 = b1/b2 = c1/c2

Number of solutions and consistency

From daily life

A lock either has a key, many keys, or none. A pair of equations is called consistent if it has at least one solution.

Simple definition

A pair is consistent if it has at least one solution (intersecting or coincident lines) and inconsistent if it has none (parallel lines). Coincident lines are also called dependent.

intersect1 solutionconsistentcoincidentinfiniteconsistentparallelnoneinconsistent

In the figure: Only parallel lines give an inconsistent pair.

Rule 1: Name the pair — Look at the ratios, then decide.

  • Easy: x − y = 2, 2x − 2y = 4: infinite, consistent (dependent).
  • Medium: x + y = 3, 2x + 2y = 7: parallel, inconsistent.
  • Exam-type: 2x + y = 5, x − y = 1: one solution, consistent.

Rule 2: True or false — Use the three cases to decide.

  • Easy: Every consistent pair has exactly one solution: false (coincident lines).
  • Medium: A pair with two solutions must have infinitely many.
  • Exam-type: Parallel lines: inconsistent.
Remember

Two different lines cannot have exactly two common points.

Common mistake

Calling coincident lines inconsistent: they have solutions, so they are consistent.

Memory clue

"Parallel is the only inconsistent."

Practice questions

Easy
  1. 1How many solutions does the pair 6x + y + 1 = 0, 12x + 2y + 2 = 0 have?
  2. 2How many solutions does the pair −7x − 6y − 4 = 0, 14x + 12y + 5 = 0 have?
  3. 3How many solutions does the pair −6x + 6y + 6 = 0, −18x + 18y + 18 = 0 have?
Medium
  1. 4How many solutions does the pair 3x − 2y + 5 = 0, −6x − 3y + 1 = 0 have?
  2. 5How many solutions does the pair 4x − 4y + 3 = 0, −8x + 8y − 6 = 0 have?
  3. 6How many solutions does the pair −x + y − 4 = 0, x + 6y − 7 = 0 have?
Challenging
  1. 7How many solutions does the pair x + 7y − 6 = 0, −2x − 14y + 14 = 0 have?
  2. 8A pair of linear equations that has at least one solution is called ___.
Show answers (try them yourself first)
  1. 1Infinitely many solutions
  2. 2No solution
  3. 3Infinitely many solutions
  4. 4Exactly one solution
  5. 5Infinitely many solutions
  6. 6Exactly one solution
  7. 7No solution
  8. 8consistent

Finding k for a given condition

From daily life

An unknown number k decides whether two lines cross, run parallel or merge. Match k to the case you want.

Simple definition

Set up the condition: unique solution: a₁/a₂ ≠ b₁/b₂. No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Infinitely many: a₁/a₂ = b₁/b₂ = c₁/c₂. Solve for k.

uniquea₁/a₂ ≠ b₁/b₂nonea₁/a₂=b₁/b₂≠c₁/c₂infiniteall three equal

In the figure: Each case is one equation or inequality in k.

Rule 1: Infinitely many — Equate all three ratios.

  • Easy: x + 2y = 3, 2x + ky = 6: 1/2 = 2/k = 3/6 ⇒ k = 4.
  • Medium: 3x + y = 1, 6x + 2y = k: 3/6 = 1/2 = 1/k ⇒ k = 2.
  • Exam-type: Check by multiplying the first equation by 2.

Rule 2: No solution or unique — No solution: a-b ratios equal but c different. Unique: a-b ratios different.

  • Easy: 2x + 3y = 5, 4x + ky = 7: 2/4 = 3/k ⇒ k = 6; 5/7 ≠ 1/2 ✓ no solution.
  • Medium: 3x + 2y = 5, kx + 4y = 7: unique when 3/k ≠ 1/2, so k ≠ 6.
  • Exam-type: x + 2y = 3 and 2x + ky = 5 have a unique solution for all k ≠ 4.
Remember

The k you find for "no solution" must be checked against the c-ratio.

Common mistake

Not checking the third ratio, so "no solution" and "infinite" get mixed up.

Memory clue

"Find k from a and b; test c."

Practice questions

Easy
  1. 1For what value of k does the pair −3x + y = −5 and kx + 4y = −18 have no solution?
  2. 2For what value of k does the pair −4x − 7y = −6 and −12x + ky = −22 have no solution?
  3. 3For what value of k does the pair −5x − 5y = 2 and kx − 10y = 7 have no solution?
Medium
  1. 4For what value of k does the pair −5x − y = 7 and kx + 3y = −19 have no solution?
  2. 5For what value of k does the pair 6x + 4y = 6 and kx + 8y = 15 have no solution?
  3. 6For what value of k does the pair 5x − 6y = 4 and kx + 12y = −6 have no solution?
Challenging
  1. 7Find the value of k for which the pair 5x − 6y = 3 and 20x + ky = 12 has infinitely many solutions.
  2. 8For what value of k does the pair −3x − y = −3 and kx + 3y = 6 have no solution?
Show answers (try them yourself first)
  1. 1−12
  2. 2−21
  3. 3−10
  4. 415
  5. 512
  6. 6−10
  7. 7−24
  8. 89

The solution is the point of intersection

From daily life

Two paths crossing at a spot: that spot belongs to both paths. The crossing point of two lines satisfies both equations.

Simple definition

To solve a pair graphically, draw both lines. The coordinates of the point where they meet are the solution (x, y). Verify by putting them in both equations.

-2-1123456789-112345xy(3, 2)x + y = 5x − y = 1

In the figure: x + y = 5 and x − y = 1 meet at (3, 2), so x = 3, y = 2.

Live figure

The solution is the point of intersection

In the app you can move this figure with your finger.

Open in the app

Rule 1: Read the point — Check by substitution in both.

  • Easy: x + y = 4, x − y = 2: point (3, 1).
  • Medium: x + 2y = 5, x − y = −1: point (1, 2): 1 + 4 = 5 ✓, 1 − 2 = −1 ✓.
  • Exam-type: y = 2x and x + y = 6: point (2, 4).

Rule 2: Plot carefully — Use at least two points for each line, three to be safe.

  • Easy: For x + y = 5: (0, 5), (5, 0), (2, 3).
  • Medium: For x − y = 1: (1, 0), (0, −1), (3, 2).
  • Exam-type: The third point must also lie on the line.
Remember

If your three points are not in a line, one of them is wrong.

Common mistake

Reading the solution as (y, x) instead of (x, y).

Memory clue

"Lines meet at the answer."

Practice questions

Easy
  1. 1The lines −4x − 5y = −28 and 7x − 6y = −10 intersect at which point?
  2. 2The solution of a pair of linear equations in x and y is represented graphically by the ___ of the two lines.
  3. 3The lines −3x − y = −20 and −3x − 7y = −14 intersect at which point?
Medium
  1. 4The lines −3x + y = −4 and 4x + 4y = 16 intersect at which point?
  2. 5The graphs of −x + 5y = −23 and 7x + 5y = 1 intersect at the point (x, y). Find the value of xy.
  3. 6The graphs of −5x − 7y = 39 and −6x − 7y = 44 intersect at the point (x, y). Find the value of x + y.
Challenging
  1. 7The lines −3x + y = −18 and −3x − 6y = 3 meet at a point P. The line kx − 2y = 36 passes through P. Find k.
  2. 8The lines −2x + 2y = −14 and x + 7y = −17 meet at a point P. The line kx + 2y = 2 passes through P. Find k.
Show answers (try them yourself first)
  1. 1(2, 4)
  2. 2point of intersection
  3. 3(7, −1)
  4. 4(2, 2)
  5. 5−12
  6. 6−7
  7. 76
  8. 82

Intercepts and forming equations

From daily life

Where a line cuts the axes gives two quick points. And word stories turn into equations to draw.

Simple definition

The x-intercept is found by putting y = 0 and the y-intercept by putting x = 0. Turn a situation into two equations by naming the unknowns x and y.

Formula: x-intercept: y = 0; y-intercept: x = 0

-11234-112xy(3, 0)(0, 2)2x + 3y = 6

In the figure: 2x + 3y = 6 cuts the axes at (3, 0) and (0, 2).

Rule 1: Find the intercepts — Put each variable as 0 in turn.

  • Easy: x + y = 5: (5, 0) and (0, 5).
  • Medium: 3x − 2y = 6: (2, 0) and (0, −3).
  • Exam-type: 4x + 2y = 8: (2, 0) and (0, 4).

Rule 2: Form the equations — Let x and y be the unknown quantities; write each statement as an equation.

  • Easy: Two numbers add to 10 and differ by 2: x + y = 10, x − y = 2.
  • Medium: 2 pens and 1 pencil cost 11; 1 pen and 1 pencil cost 7: 2x + y = 11, x + y = 7 (pen 4, pencil 3).
  • Exam-type: Father is 3 times the son and the sum is 48: x = 3y, x + y = 48 (son 12).
Remember

Always say what x and y stand for before writing the equations.

Common mistake

Putting x = 0 to find the x-intercept (it is y = 0).

Memory clue

"x-intercept: y is zero."

Practice questions

Easy
  1. 1To draw the graph of a linear equation in two variables, the least number of points we need to plot is ___.
  2. 2The sum of two numbers is 18 and their difference is 13. Which pair of equations represents this (x is the bigger number, y the smaller)?
  3. 3The sum of two numbers is 25 and their difference is 15. Which pair of equations represents this (x is the bigger number, y the smaller)?
Medium
  1. 4The sum of two numbers is 13 and their difference is 3. Which pair of equations represents this (x is the bigger number, y the smaller)?
  2. 5One number x is 9 more than twice another number y, and their sum is 34. Which pair of equations represents this?
  3. 6One number x is 2 more than three times another number y, and their sum is 28. Which pair of equations represents this?
Challenging
  1. 7Find the area of the triangle formed by the line 6x + 5y = 30 and the two coordinate axes.
  2. 8Find the area of the triangle formed by the line x + 5y = 5 and the two coordinate axes.
Show answers (try them yourself first)
  1. 12
  2. 2x + y = 18 and x − y = 13
  3. 3x + y = 25 and x − y = 15
  4. 4x + y = 13 and x − y = 3
  5. 5x − 2y = 9 and x + y = 34
  6. 6x − 3y = 2 and x + y = 28
  7. 715
  8. 85/2