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NCERT Solutions · Class 10 · Chapter 3: Pair of Linear Equations in Two Variables

NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.1

Exercise 3.1: Graphs and the ratios of coefficients. For \(a_1x + b_1y + c_1 = 0\), \(a_2x + b_2y + c_2 = 0\): \(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}\) means intersecting lines (one solution); \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\) coincident lines (infinitely many); \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}\) parallel lines (none).

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Exercise 3.1 questions and solutions

Exercise 3.1, Question 1

Form the pair of equations and find the solution graphically.
(i) 10 students took part in a quiz; there were 4 more girls than boys. Find the numbers of boys and girls.
Show solution
  1. Let \(x\) = girls, \(y\) = boys: \(x + y = 10\) and \(x - y = 4\).
  2. Plot both lines (e.g. through \((10, 0), (0, 10)\) and \((4, 0), (7, 3)\)); they meet at \((7, 3)\).
Answer: 7 girls and 3 boys
(ii) 5 pencils and 7 pens cost ₹50; 7 pencils and 5 pens cost ₹46. Find the cost of one pencil and one pen.
Show solution
  1. Let a pencil cost ₹\(x\) and a pen ₹\(y\): \(5x + 7y = 50\), \(7x + 5y = 46\).
  2. The lines meet at \((3, 5)\).
Answer: Pencil ₹3, pen ₹5

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Exercise 3.1, Question 2

By comparing ratios, decide whether the lines intersect, are parallel or coincide.
(i) \(5x - 4y + 8 = 0,\ 7x + 6y - 9 = 0\)
Show solution
  1. Compare \(\dfrac{a_1}{a_2}\), \(\dfrac{b_1}{b_2}\) and \(\dfrac{c_1}{c_2}\).
  2. \(\dfrac57 \ne \dfrac{-4}{6}\).
Answer: Intersect at one point
(ii) \(9x + 3y + 12 = 0,\ 18x + 6y + 24 = 0\)
Show solution
  1. Compare \(\dfrac{a_1}{a_2}\), \(\dfrac{b_1}{b_2}\) and \(\dfrac{c_1}{c_2}\).
  2. \[\begin{aligned}\dfrac{9}{18} &= \dfrac36 \\ &= \dfrac{12}{24} \\ &= \dfrac12\end{aligned}\]
Answer: Coincident
(iii) \(6x - 3y + 10 = 0,\ 2x - y + 9 = 0\)
Show solution
  1. Compare \(\dfrac{a_1}{a_2}\), \(\dfrac{b_1}{b_2}\) and \(\dfrac{c_1}{c_2}\).
  2. \(\dfrac62 = \dfrac{-3}{-1} = 3\) but \(\dfrac{10}{9} \ne 3\).
Answer: Parallel

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Exercise 3.1, Question 3

By comparing ratios, decide whether each pair is consistent or inconsistent.
(i) \(3x + 2y = 5,\ 2x - 3y = 7\)
Show solution
  1. Compare \(\dfrac{a_1}{a_2}\), \(\dfrac{b_1}{b_2}\) and \(\dfrac{c_1}{c_2}\).
  2. \(\dfrac32 \ne \dfrac{2}{-3}\): unique solution.
Answer: Consistent
(ii) \(2x - 3y = 8,\ 4x - 6y = 9\)
Show solution
  1. Compare \(\dfrac{a_1}{a_2}\), \(\dfrac{b_1}{b_2}\) and \(\dfrac{c_1}{c_2}\).
  2. \(\dfrac24 = \dfrac{-3}{-6} \ne \dfrac89\): parallel.
Answer: Inconsistent
(iii) \(\tfrac32x + \tfrac53y = 7,\ 9x - 10y = 14\)
Show solution
  1. Compare \(\dfrac{a_1}{a_2}\), \(\dfrac{b_1}{b_2}\) and \(\dfrac{c_1}{c_2}\).
  2. \(\dfrac{3/2}{9} = \dfrac16\), \(\dfrac{5/3}{-10} = -\dfrac16\): not equal, unique solution.
Answer: Consistent
(iv) \(5x - 3y = 11,\ -10x + 6y = -22\)
Show solution
  1. Compare \(\dfrac{a_1}{a_2}\), \(\dfrac{b_1}{b_2}\) and \(\dfrac{c_1}{c_2}\).
  2. All three ratios equal \(-\tfrac12\): coincident lines.
Answer: Consistent (infinitely many solutions)
(v) \(\tfrac43x + 2y = 8,\ 2x + 3y = 12\)
Show solution
  1. Compare \(\dfrac{a_1}{a_2}\), \(\dfrac{b_1}{b_2}\) and \(\dfrac{c_1}{c_2}\).
  2. All three ratios equal \(\tfrac23\): coincident lines.
Answer: Consistent (infinitely many solutions)

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Exercise 3.1, Question 4

Which pairs are consistent? Solve those graphically.
(i) \(x + y = 5,\ 2x + 2y = 10\)
Show solution
  1. Ratios all \(\tfrac12\): the same line.
Answer: Consistent; every point on \(x + y = 5\), e.g. \((0, 5), (5, 0), (2, 3)\)
(ii) \(x - y = 8,\ 3x - 3y = 16\)
Show solution
  1. \(\dfrac13 = \dfrac13 \ne \dfrac{8}{16}\): parallel lines.
Answer: Inconsistent
(iii) \(2x + y - 6 = 0,\ 4x - 2y - 4 = 0\)
Show solution
  1. \(\dfrac24 \ne \dfrac{1}{-2}\): the lines meet once.
  2. From the graphs (or \(y = 6 - 2x\) into the second): \(x = 2, y = 2\).
Answer: Consistent; \(x = 2,\ y = 2\)
(iv) \(2x - 2y - 2 = 0,\ 4x - 4y - 5 = 0\)
Show solution
  1. \[\dfrac24 = \dfrac{-2}{-4} \ne \dfrac{-2}{-5}\]: parallel lines.
Answer: Inconsistent

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Exercise 3.1, Question 5

Half the perimeter of a rectangular garden is 36 m, and its length is 4 m more than its width. Find its dimensions.
Show solution
  1. Let the length be \(x\) m and the width \(y\) m: \(x + y = 36\) and \(x - y = 4\).
  2. The lines meet at \(x = 20, y = 16\).
Answer: Length 20 m, width 16 m

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Exercise 3.1, Question 6

Given \(2x + 3y - 8 = 0\), write another linear equation so that the pair is:
(i) intersecting lines
Show solution
  1. Choose \(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}\), for example \(3x - 2y - 7 = 0\).
Answer: e.g. \(3x - 2y - 7 = 0\)
(ii) parallel lines
Show solution
  1. Keep the ratio of \(x\) and \(y\) coefficients, change the constant: e.g. \(4x + 6y - 9 = 0\).
Answer: e.g. \(4x + 6y - 9 = 0\)
(iii) coincident lines
Show solution
  1. Multiply the whole equation by a constant: e.g. \(6x + 9y - 24 = 0\).
Answer: e.g. \(6x + 9y - 24 = 0\)

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Exercise 3.1, Question 7

Draw \(x - y + 1 = 0\) and \(3x + 2y - 12 = 0\). Find the vertices of the triangle they form with the x-axis.
Show solution
  1. The lines meet where \(y = x + 1\) and \(3x + 2(x + 1) = 12\): \(x = 2, y = 3\).
  2. They cut the x-axis (\(y = 0\)) at \((-1, 0)\) and \((4, 0)\).
Answer: \((2, 3),\ (-1, 0),\ (4, 0)\)

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