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NCERT Class 10 Mathematics Exercise 3.1 Solutions: Pair of Linear Equations in Two Variables

Step-by-step working for every question of Exercise 3.1 in NCERT's Class 10 Mathematics book, chapter “Pair of Linear Equations in Two Variables”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.

1. Ex 3.1 Q1

   Form the equations and solve graphically: (i) 10 students took a quiz and there were 4 more girls than boys — how many boys and girls? (ii) 5 pencils and 7 pens cost ₹50; 7 pencils and 5 pens cost ₹46 — cost of one pencil and one pen?

   1. (i) Let the number of boys be x and girls be y.
   2. Total students are 10, so x + y = 10.
   3. There are 4 more girls than boys, so y − x = 4.
   4. Draw both lines: x + y = 10 passes through (10, 0) and (0, 10); y − x = 4 passes through (0, 4) and (−4, 0). They cross at (3, 7), so x = 3 and y = 7.
   5. (ii) Let the cost of one pencil be x and one pen be y.
   6. 5 pencils and 7 pens cost ₹50, so 5x + 7y = 50.
   7. 7 pencils and 5 pens cost ₹46, so 7x + 5y = 46.
   8. Draw both lines (5x + 7y = 50 through (10, 0) and (3, 5); 7x + 5y = 46 through (3, 5) and (8, −2)). They cross at (3, 5), so x = 3 and y = 5.

   Answer: (i) 3 boys, 7 girls; (ii) pencil ₹3, pen ₹5

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2. Ex 3.1 Q2

   Compare a₁/a₂, b₁/b₂, c₁/c₂ to say whether the lines intersect, are parallel or coincide: (i) 5x − 4y + 8 = 0, 7x + 6y − 9 = 0 (ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0 (iii) 6x − 3y + 10 = 0, 2x − y + 9 = 0.

   1. (i) For 5x − 4y + 8 = 0 and 7x + 6y − 9 = 0, a₁/a₂ = 5/7, b₁/b₂ = −4/6 = −2/3, c₁/c₂ = 8/−9 = −8/9.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   2. (i) Since 5/7 ≠ −2/3 ≠ −8/9, the ratios are all different, so the lines intersect at a unique point.
   3. (ii) For 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0, a₁/a₂ = 9/18 = 1/2, b₁/b₂ = 3/6 = 1/2, c₁/c₂ = 12/24 = 1/2.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   4. (ii) Since 1/2 = 1/2 = 1/2, the ratios are equal, so the lines coincide.
   5. (iii) For 6x − 3y + 10 = 0 and 2x − y + 9 = 0, a₁/a₂ = 6/2 = 3, b₁/b₂ = −3/−1 = 3, c₁/c₂ = 10/9.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   6. (iii) Since 3 = 3 ≠ 10/9, the first two ratios are equal but the third is different, so the lines are parallel.

   Answer: (i) intersecting; (ii) coincident; (iii) parallel

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3. Ex 3.1 Q3

   Compare the ratios to say whether each pair is consistent or inconsistent: (i) 3x + 2y = 5; 2x − 3y = 7 (ii) 2x − 3y = 8; 4x − 6y = 9 (iii) (3/2)x + (5/3)y = 7; 9x − 10y = 14 (iv) 5x − 3y = 11; −10x + 6y = −22 (v) (4/3)x + 2y = 8; 2x + 3y = 12.

   1. (i) For 3x + 2y = 5 and 2x − 3y = 7, a₁/a₂ = 3/2 and b₁/b₂ = 2/−3 = −2/3.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   2. (i) Since 3/2 ≠ −2/3, the lines intersect at a unique point, so the pair is consistent.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   3. (ii) For 2x − 3y = 8 and 4x − 6y = 9, a₁/a₂ = 2/4 = 1/2 and b₁/b₂ = −3/−6 = 1/2.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   4. (ii) Since a₁/a₂ = b₁/b₂ = 1/2 but c₁/c₂ = 8/9, the lines are parallel and do not intersect, so the pair is inconsistent.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   5. (iii) For (3/2)x + (5/3)y = 7 and 9x − 10y = 14, a₁/a₂ = (3/2)/9 = 1/6 and b₁/b₂ = (5/3)/−10 = −1/6.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   6. (iii) Since 1/6 ≠ −1/6, the lines intersect at a unique point, so the pair is consistent.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   7. (iv) For 5x − 3y = 11 and −10x + 6y = −22, a₁/a₂ = 5/−10 = −1/2 and b₁/b₂ = −3/6 = −1/2.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   8. (iv) Since a₁/a₂ = b₁/b₂ = c₁/c₂ = −1/2, the lines are coincident and have infinitely many solutions, so the pair is consistent.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   9. (v) For (4/3)x + 2y = 8 and 2x + 3y = 12, a₁/a₂ = (4/3)/2 = 2/3 and b₁/b₂ = 2/3.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   10. (v) Since a₁/a₂ = b₁/b₂ = c₁/c₂ = 2/3 (c₁/c₂ = 8/12 = 2/3), the lines are coincident, so the pair is consistent.
       why?

       For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   Answer: (i) consistent; (ii) inconsistent; (iii) consistent; (iv) consistent; (v) consistent

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4. Ex 3.1 Q4

   Consistent or inconsistent? If consistent, solve graphically: (i) x + y = 5, 2x + 2y = 10 (ii) x − y = 8, 3x − 3y = 16 (iii) 2x + y − 6 = 0, 4x − 2y − 4 = 0 (iv) 2x − 2y − 2 = 0, 4x − 4y − 5 = 0.

   1. (i) The equations are x + y = 5 and 2x + 2y = 10.
   2. Dividing the second equation by 2 gives x + y = 5, which is identical to the first equation.
   3. Since the lines coincide, the system is consistent with infinitely many solutions.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   4. (ii) The equations are x − y = 8 and 3x − 3y = 16.
   5. Dividing the second equation by 3 gives x − y = 16/3.
   6. Since 8 is not equal to 16/3, the lines are parallel and distinct, making the system inconsistent.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   7. (iii) The equations are 2x + y = 6 and 4x − 2y = 4.
   8. Solving the first for y gives y = 6 − 2x.
   9. Substituting into the second gives 4x − 2(6 − 2x) = 4, which simplifies to 8x − 12 = 4.
      why?

      Putting the known values in place of the letters turns the formula or equation into plain arithmetic — the letters stood for exactly these values.

   10. Solving for x gives 8x = 16, so x = 2.
   11. Substituting x = 2 back gives y = 6 − 4 = 2, so the system is consistent.
       why?

       For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   12. (iv) The equations are 2x − 2y = 2 and 4x − 4y = 5.
   13. Dividing the second equation by 2 gives 2x − 2y = 2.5.
   14. Since 2 is not equal to 2.5, the lines are parallel and distinct, making the system inconsistent.
       why?

       For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   Answer: (i) consistent; (ii) inconsistent; (iii) consistent, x = 2, y = 2; (iv) inconsistent

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5. Ex 3.1 Q5

   Half the perimeter of a rectangular garden is 36 m and its length is 4 m more than its width. Find its dimensions.

   1. Let the width of the garden be x m and the length be y m.
   2. Half the perimeter is given as 36 m, so x + y = 36.
   3. The length is 4 m more than the width, so y = x + 4.
   4. Substitute y = x + 4 into the first equation to get x + (x + 4) = 36.
      why?

      Putting the known values in place of the letters turns the formula or equation into plain arithmetic — the letters stood for exactly these values.

   5. Simplify the equation to 2x + 4 = 36.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   6. Subtract 4 from both sides to get 2x = 32.
      why?

      Doing the same thing to both sides of an equation (adding, subtracting, multiplying or dividing by the same non-zero number) keeps the two sides equal, so the equation stays true while the unknown is left on its own.

   7. Divide by 2 to find x = 16.
   8. Calculate y using y = 16 + 4, which gives y = 20.

   Answer: length 20 m, width 16 m

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6. Ex 3.1 Q6

   For 2x + 3y − 8 = 0, write another linear equation so that the pair gives (i) intersecting lines (ii) parallel lines (iii) coincident lines.

   1. Compare a₁/a₂, b₁/b₂ and c₁/c₂ for 2x + 3y − 8 = 0 and the new line a₂x + b₂y + c₂ = 0.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   2. (i) Intersecting lines need a₁/a₂ ≠ b₁/b₂. Take 3x + 2y − 7 = 0: 2/3 ≠ 3/2, so the lines intersect.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   3. (ii) Parallel lines need a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Take 4x + 6y + 3 = 0: 2/4 = 3/6 = 1/2, but c₁/c₂ = −8/3 ≠ 1/2, so the lines are parallel.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   4. (iii) Coincident lines need all three ratios equal. Take 4x + 6y − 16 = 0 (the same equation multiplied by 2): 2/4 = 3/6 = −8/−16 = 1/2.
      why?

      For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives one solution (the lines cross); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none (parallel lines); all three ratios equal gives infinitely many (the same line).

   Answer: (i) 3x + 2y − 7 = 0; (ii) 4x + 6y + 3 = 0; (iii) 4x + 6y − 16 = 0 (other answers following the same ratio rules are also correct)

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7. Ex 3.1 Q7

   Draw x − y + 1 = 0 and 3x + 2y − 12 = 0. Find the vertices of the triangle these lines form with the x-axis.

   1. Find the x-intercept of x − y + 1 = 0 by setting y = 0, which gives x = −1, so the point is (−1, 0).
   2. Find the x-intercept of 3x + 2y − 12 = 0 by setting y = 0, which gives 3x = 12, so x = 4, and the point is (4, 0).
   3. Find the intersection of the two lines by solving x − y + 1 = 0 and 3x + 2y − 12 = 0 simultaneously.
   4. From the first equation, y = x + 1.
   5. Substitute y = x + 1 into the second equation: 3x + 2(x + 1) − 12 = 0.
      why?

      Putting the known values in place of the letters turns the formula or equation into plain arithmetic — the letters stood for exactly these values.

   6. Simplify to get 5x − 10 = 0, so x = 2.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   7. Substitute x = 2 into y = x + 1 to get y = 3, so the intersection point is (2, 3).
      why?

      Putting the known values in place of the letters turns the formula or equation into plain arithmetic — the letters stood for exactly these values.

   8. The vertices of the triangle formed by these lines and the x-axis are (−1, 0), (4, 0), and (2, 3).

   Answer: The vertices are (−1, 0), (4, 0), and (2, 3).

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