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CBSE · Class 9 · Mathematics

Chapter 13: Linear Equations in Two Variables

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

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Linear equations in two variables

From daily life

Pens cost ₹5 and notebooks ₹10. If you spend ₹50, many pairs of pens and notebooks work: 10 pens, or 6 pens and 2 notebooks, and so on.

Simple definition

An equation ax + by + c = 0 (a, b not both 0) is a linear equation in two variables. A solution is a pair (x, y) that makes it true. It has infinitely many solutions.

Formula: ax + by + c = 0; solution (x, y)

x=0y=7x=1y=6x=2y=5x=3y=4x=4y=3x=5y=2

In the figure: For x + y = 7 the pairs (0, 7), (1, 6), (2, 5), (3, 4), … all work.

Rule 1: Is it a solution? — Put x and y into the equation; both sides must match.

  • Easy: 2x + y = 9: (3, 3) works since 6 + 3 = 9.
  • Medium: x − y = 2: (5, 4) does not (5 − 4 = 1).
  • Exam-type: 3x + 2y = 12: (0, 6) works.

Rule 2: Find more solutions — Choose x, then solve for y.

  • Easy: x + 2y = 8: x = 2 ⇒ y = 3.
  • Medium: y = 2x + 1: x = 3 ⇒ y = 7.
  • Exam-type: Write a word problem as an equation: 5p + 10b = 50.
Remember

One equation in two variables has infinitely many solutions, not one.

Common mistake

Giving only one solution as "the" answer.

Memory clue

"Two unknowns, many pairs."

Practice questions

Easy
  1. 1Is (9, −4) a solution of 5x + y = 37?
  2. 2If x = 0 in the equation 4x + 5y = 5, what is y?
  3. 3If x = −4 in the equation 4x + 3y = −7, what is y?
Medium
  1. 4(4, 1) is a solution of (k + 3)x + 6y = 10. Find k.
  2. 5Pens cost ₹19 each and notebooks cost ₹14 each. Riya spends ₹122 on x pens and y notebooks. If she buys 2 pens, how many notebooks does she buy?
  3. 6Compare 8x + 6y + 5 = 0 with ax + by + c = 0. What are a, b and c?
Challenging
  1. 7How many solutions of 2x + 4y = 30 have both x and y as positive whole numbers?
  2. 8The equation 3x + ky = 34 has the solution x = 2, y = 4. Find k and the value of y when x = 0.
Show answers (try them yourself first)
  1. 1No
  2. 21
  3. 33
  4. 4−2
  5. 56
  6. 6a = 8, b = 6, c = 5
  7. 77
  8. 8k = 7, y = 34/7

Graph of a linear equation

From daily life

Plot each pair of x + y = 4 as a dot on graph paper and the dots line up in a straight line.

Simple definition

The graph of a linear equation in two variables is a straight line. Every point on the line is a solution and every solution is a point on the line. Two points are enough to draw it; a third checks it.

Formula: Points: set x = 0 for the y-intercept, y = 0 for the x-intercept

xy−112345−112345(0,4)(4,0)(2,2)

In the figure: The line x + y = 4 passes through (0, 4), (2, 2) and (4, 0).

Rule 1: Draw the line — Find the two intercepts, plot, join and extend both ways.

  • Easy: 2x + y = 6: (0, 6) and (3, 0).
  • Medium: x − y = 1: (0, −1) and (1, 0).
  • Exam-type: x = 3 is a vertical line; y = −2 a horizontal line.

Rule 2: Points on the line — A point lies on the line only if its coordinates satisfy the equation.

  • Easy: (1, 3) lies on x + y = 4, but (2, 3) does not.
  • Medium: Line y = 2x passes through the origin.
  • Exam-type: If (k, 2) lies on x + y = 7 then k = 5.
Remember

Axes: the x-axis has equation y = 0 and the y-axis x = 0.

Common mistake

Joining the points with a curve or stopping the line at the plotted points.

Memory clue

"Two points make a line, a third proves it."

Practice questions

Easy
  1. 1Where does the line 3x + 7y = 6 cut the x-axis?
  2. 2Where does the line 7x + 5y = 40 cut the y-axis?
  3. 3Which point lies on the line x − 2y = 1?
Medium
  1. 4A line is parallel to the x-axis and passes through (−2, 5). What is its equation?
  2. 5The points (0, −3), (−2, −13) and (−3, −18) lie on a line. What is y when x = 9?
  3. 6The line 3x + 5y = 45 and the two axes form a triangle. What is the area of the triangle (square units)?
Challenging
  1. 7A line cuts the x-axis at (3, 0) and the y-axis at (0, 4). What is the distance between these two points?
  2. 8A line makes intercepts 4 on the x-axis and 14 on the y-axis. What are the coordinates of the midpoint of the segment joining its intercepts on the axes?
Show answers (try them yourself first)
  1. 1(2, 0)
  2. 2(0, 8)
  3. 3(1, 0)
  4. 4y = 5
  5. 542
  6. 6135/2
  7. 75
  8. 8(2, 7)

Slope and slope-intercept form

From daily life

A ramp rises 1 m over 4 m of ground; its steepness is 1/4. A steeper ramp has a bigger number.

Simple definition

The slope m = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁). In y = mx + c, m is the slope and c the y-intercept. Positive m rises, negative m falls, m = 0 is horizontal.

Formula: m = (y₂ − y₁)/(x₂ − x₁); y = mx + c

xy−2−1123−2−112345(0,1)(1,3)

In the figure: From (0, 1) go 1 right and 2 up to (1, 3): rise 2, run 1, slope 2, so y = 2x + 1.

Rule 1: Find the slope — Use two points, rise over run.

  • Easy: (1, 2) and (3, 8): m = 6/2 = 3.
  • Medium: (0, 5) and (5, 0): m = −1.
  • Exam-type: 2x + 3y = 6 ⇒ y = −(2/3)x + 2, slope −2/3.

Rule 2: Read y = mx + c — Rearrange so y is alone; then m and c can be read off.

  • Easy: y = 4x − 3: slope 4, y-intercept −3.
  • Medium: Parallel lines have equal slopes.
  • Exam-type: Slope 1/2 and passes (0, 3): y = x/2 + 3.
Remember

Slope is the same everywhere on a straight line.

Common mistake

Writing run over rise (x over y) instead of rise over run.

Memory clue

"Rise over run: up first."

Practice questions

Easy
  1. 1What is the slope of the line through (−1, 2) and (0, 1)?
  2. 2For the line y = −2x + 1, what is the y-intercept?
  3. 3What is the slope of the line 6x + 3y = 9?
Medium
  1. 4A line has slope 2. What is the slope of any line parallel to it?
  2. 5A line has slope −5/3. What is the slope of a line perpendicular to it?
  3. 6Which is the equation of the line with slope −2 and y-intercept 6?
Challenging
  1. 7A line has slope 2 and passes through (4, 4). What is its y-intercept?
  2. 8The line through (1, −2) and (3, k) has slope 2. Find k.
Show answers (try them yourself first)
  1. 1−1
  2. 21
  3. 3−2
  4. 42
  5. 53/5
  6. 6y = −2x + 6
  7. 7−4
  8. 82

Solving a pair algebraically

From daily life

Two friends share the cost: together ₹10 and one pays ₹2 more. Two clues, two unknowns; both clues together fix the amounts.

Simple definition

A pair of linear equations has a common solution. By substitution write one variable in terms of the other and put it in the second. By elimination add or subtract the equations to remove one variable.

Formula: Substitution or elimination, then check in both equations

x + y = 10x − y = 22x = 12 ⇒ x = 6y = 10 − 6 = 4

In the figure: Adding x + y = 10 and x − y = 2 removes y: 2x = 12, x = 6, then y = 4.

Rule 1: Substitution — Express one variable from the simpler equation.

  • Easy: y = x + 1 and x + y = 7: 2x + 1 = 7, x = 3, y = 4.
  • Medium: x = 2y and x + y = 9: 3y = 9, y = 3, x = 6.
  • Exam-type: Always check: (3, 4) satisfies both.

Rule 2: Elimination — Make a coefficient equal, then add or subtract.

  • Easy: 2x + y = 8 and x − y = 1: add ⇒ 3x = 9, x = 3, y = 2.
  • Medium: x + 2y = 7 and x + y = 4: subtract ⇒ y = 3, x = 1.
  • Exam-type: 3x + 2y = 12 and x + y = 5: ×2 on the second ⇒ x = 2, y = 3.
Remember

Both methods give the same answer; choose the one with less work.

Common mistake

Finding x and forgetting to find y, or not checking in both equations.

Memory clue

"Remove one, find one, find the other."

Practice questions

Easy
  1. 1Is x = −1, y = −1 the solution of both −3x + 4y = −1 and −3x − y = 4?
  2. 2Solve x + y = 23 and x − y = 5. What is x?
  3. 3Solve the pair x + 2y = 10 and −3x + 2y = 2. What is the value of x?
Medium
  1. 4Solve the pair −x + 3y = 14 and −x + 4y = 19. What is the value of y?
  2. 5Solve 5x − 2y = −23 and −x + 5y = 23. What is x + y?
  3. 63 pencils and 2 erasers cost ₹45; 5 pencils and 4 erasers cost ₹83. What is the cost of one pencil (in ₹)?
Challenging
  1. 7A father is 31 years older than his son. The sum of their ages is 63 years. How old is the son?
  2. 8The sum of the digits of a two-digit number is 14. The number is 36 more than the number formed by reversing its digits. What is the number?
Show answers (try them yourself first)
  1. 1Yes
  2. 214
  3. 32
  4. 45
  5. 51
  6. 67
  7. 716
  8. 895

Graphical method and consistency

From daily life

Two roads either cross at one junction, never meet if parallel, or are the same road. Two lines behave in just these three ways.

Simple definition

Draw both lines. Intersect at a point: exactly one solution (consistent). Parallel: no solution (inconsistent). Coincident: infinitely many solutions (consistent). The intersection point is the solution.

Formula: a₁/a₂ ≠ b₁/b₂ one solution; = but c differs: none; all equal: infinite

xy−1123456−2−1123456(4,2)

In the figure: x + y = 6 and x − y = 2 cross at (4, 2), which is the solution of the pair.

Rule 1: Count solutions — Compare slopes and intercepts.

  • Easy: x + y = 4 and x + y = 6: parallel, no solution.
  • Medium: x + y = 4 and 2x + 2y = 8: same line, infinitely many.
  • Exam-type: x + y = 4 and x − y = 0: one solution (2, 2).

Rule 2: Read the intersection — Check by putting the point into both equations.

  • Easy: Lines meet at (4, 2): x = 4, y = 2.
  • Medium: Equal slope, different intercepts: never meet.
  • Exam-type: A consistent pair has at least one solution.
Remember

Graphs give approximate answers when the intersection is not on a grid point; algebra gives exact ones.

Common mistake

Saying parallel lines have one solution, or that no intersection means a mistake.

Memory clue

"Cross: one. Parallel: none. Same: many."

Practice questions

Easy
  1. 1How many solutions does the pair x + 4y = 7 and 2x + y = 9 have?
  2. 2The lines 4x + 2y = −14 and 4x − y = −11 are drawn on the same graph. At which point do they cross?
  3. 3The lines 3x + y = 28 and 2x − 4y = 0 are drawn on the same graph. At which point do they cross?
Medium
  1. 4For which value of p does the pair x + 2y = 3 and p x + 8y = 15 have no solution?
  2. 5For which value of p do 3x + 3y = 6 and 9x + 9y = p have infinitely many solutions?
  3. 6The pair 4x + 5y = 12 and 8x + 10y = 23 is given. Without solving, how many solutions does it have?
Challenging
  1. 7If x + 2y = 6 and p x + q y = 18 represent the same line, find p + q.
  2. 8The line 3x + 6y = 18 together with the x-axis and the y-axis encloses a triangle. What is its area?
Show answers (try them yourself first)
  1. 1exactly one solution
  2. 2(−3, −1)
  3. 3(8, 4)
  4. 44
  5. 518
  6. 6no solution
  7. 79
  8. 89
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Graphical method and consistency

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