Linear equations in two variables: Questions and Answers

81 linear equations in two variables questions solved step by step. Open a question for the full working, the marks for each step and exam practice.

• Radha stops at a fruit cart. The price tags say that mangoes cost ₹60 per kg and bananas cost ₹50 per kg. She buys some kilograms of mangoes and some kilograms of bananas. Altogether, she pays ₹280. Can we express this situation as an equation?Answer: Yes. Let x = kg of mangoes and y = kg of bananas. Then 60x + 50y = 280, a linear equation in two variables.
• Write each of the following equations in standard form, ax + by + c = 0, and indicate the values of a, b and c in each case:Answer: (i) 5x + 2y − 2.7 = 0: a = 5, b = 2, c = −2.7 (ii) 12x − 32y − 4 = 0: a = 12, b = −32, c = −4 (iii) 1.5x − 1.3y − 4 = 0: a = 1.5, b = −1.3, c = −4 (iv) 0×x + 3y − 5 = 0: a = 0, b = 3, c = −5
• Write a linear equation in two variables in which a = 3, b = 0 and c = −15.Answer: 3x + 0×y − 15 = 0, that is, 3x − 15 = 0 (or 15x − 1 = 0).
• Complete the following table after expressing the given linear equations in standard form.
  

  Linear equation in two variables	Standard Form	Coefficient of x	Coefficient of y	Constant term
  y − 15 = √2x
  3y − 2x = 0
  5x = 3y
  x = 8
  3y = 1

  Answer: √2x − y + 15 = 0 (√2, −1, 15); −2x + 3y + 0 = 0 (−2, 3, 0); 5x − 3y + 0 = 0 (5, −3, 0); x + 0×y − 8 = 0 (1, 0, −8); 0×x + 3y − 1 = 0 (0, 3, −1). Numbers in brackets: coefficient of x, coefficient of y, constant term.
• (i) The cost of a notebook is twice the cost of a pen. Consider the cost of a notebook to be ₹t and that of a pen to be ₹p. Charlie wrote t = 2p, whereas Meera wrote p = 2t. Which of these two representations is correct?
  (ii) In a one-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian batsmen together scored 176 runs. Manisha expressed this situation as x + y = 176, where the number of runs scored by one batsman is x, and the number of runs scored by the other is y. Is this a correct representation?Answer: (i) Charlie is correct: t = 2p. (ii) Yes, x + y = 176 is correct.
• Let us consider the equation 3x + 2y = 12. What about the ordered pair (1, 3)?Answer: (1, 3) is not a solution: 3 × 1 + 2 × 3 = 9 ≠ 12.
• Are (1, 92), (0, 6), (4, 1) and (12, 214) solutions of the equation 3x + 2y = 12?
  Can you find any other solution? How many solutions can you find?Answer: (1, 92), (0, 6) and (12, 214) are solutions; (4, 1) is not (it gives 14). Other solutions: (2, 3), (4, 0), (−2, 9) … There are infinitely many solutions.
• Consider all the solutions of a linear equation in two variables, expressed as ordered pairs. What do we get when we plot the points corresponding to these ordered pairs?Answer: The points all lie on one straight line. That line is the graph of the equation: every solution is a point on it, and every point on it is a solution.
• Find any two solutions of 2x + y = 7. Draw the graph of this equation. Use the graph to find 4 more solutions.Answer: Two solutions: (0, 7) and (72, 0). Joining them gives the graph. From the graph, four more: (1, 5), (2, 3), (3, 1), (32, 4).
• Find any two solutions for each of the following equations and draw their graphs:Answer: (i) 4x + 3y = 12: (0, 4) and (3, 0). (ii) 2x + 5y = 0: (0, 0), (1, −25) and (5, −2); this line passes through the origin.
• In the equations shown below, a and b are unknown numbers.
  3ax + 4y = −2
  2x + by = 14.
  If (−3, 4) is the solution of both equations, find the values of a and b.Answer: a = 2, b = 5.
• Four friends — Ranju, Meena, Farhan, and Toshi — are solving problems.
  Ranju: I noticed something! If c = 0 in the standard form of a line ax + by + c = 0, then the line must pass through the origin. Look, if I substitute x = 0 and y = 0, in equation ax + by = 0, the equation is satisfied. So, the origin lies on the line! We can also say that the line passes through the origin.
  Farhan: Let us try for the equation 2x + 3y = 0. Here a = 2, b = 3 but c = 0. If we substitute x = 0, then we get 3y = 0 or y = 0. This means (0, 0) lies on the line. So yes, this line passes through the origin.
  Toshi: Suppose our equation has b = c = 0, say, 5x = 0. This becomes x = 0 which is the equation of the y-axis. And the y-axis passes through the origin.
  Meena: And if we take a = c = 0, say, 7y = 0, that means y = 0, which is the equation of the x-axis that also passes through the origin.
  So, they conclude: Whenever c = 0, the line ax + by + c = 0 will always pass through the origin, irrespective of the values of a or b. Do you agree with them?Answer: Yes. If c = 0 the equation is ax + by = 0, and x = 0, y = 0 gives a(0) + b(0) = 0, which is always true. So (0, 0) is always on the line. (The converse also holds: a line through the origin must have c = 0.)
• Verify if the ordered pair (4, 3) is a solution of 5x − 6y = 2. Explain your reasoning.Answer: Yes. 5(4) − 6(3) = 20 − 18 = 2, which equals the right-hand side.
• Find any two solutions for each of the following equations:Answer: (i) (0, −7) and (3, 0) (ii) (1, 1) and (4, −1). (Many other answers are possible.)
• In the equations given below, m and n are unknown constants: 2mx + 3y = 7; 4x + ny = −10. If (2, −1) is the solution of both equations, find the values of m and n.Answer: m = 52, n = 18.
• Find two solutions which lie in different quadrants for each of the following linear equations. Identify the quadrants in which the points lie.
  Verify your solutions by representing the linear equations on a graph paper.Answer: (i) (2, −1) in IV, (−1, 4) in II (ii) (2, 1) in I, (−1, −4) in III (iii) (1, 4) in I, (−2, −1) in III (iv) (1, −4) in IV, (−2, 1) in II.
• Consider the graph of the equation 3x − 7y = 21 shown below. Does the point C (2, 3) lie on the line? Does it satisfy the equation? Can points that do not lie on the line satisfy the equation?Answer: No, C(2, 3) is not on the line; no, it does not satisfy the equation (3(2) − 7(3) = −15 ≠ 21); and no, a point off the line can never satisfy the equation.
• State whether the following sentences are True or False. Justify your answer.Answer: (i) False (ii) False (iii) False (iv) True (v) True (vi) False.
• (i) Compare the solutions of the equations 3x + 4y = 7 and 6x + 8y = 14. Argue that they have the same set of solutions, that is, every solution of one is also a solution of the other.
  (ii) Show that the equations ax + by = c and kax + kby = kc, with k ≠ 0, have the same set of solutions.Answer: (i) 6x + 8y = 14 is just 3x + 4y = 7 multiplied by 2, so a pair satisfies one exactly when it satisfies the other. (ii) Multiplying by k ≠ 0 keeps true equations true, and dividing by k undoes it, so the solution sets are equal.
• Have you ever walked or cycled up a steep hill? How did it feel compared to walking on a flat road?Answer: Going up a steep hill is much harder than walking on a flat road: you have to push harder because you rise a lot for every step forward. The steeper the hill, the more effort. In maths this steepness is called the slope (or gradient).
• 1. Consider the point on the line AB whose x-coordinate is 8. What is the y-coordinate of this point? Express this as an ordered pair.
  2. What is the x-coordinate of the point on the line AB whose y-coordinate is 6? Express this as an ordered pair.Answer: 1. (8, 9) 2. (6, 6). The line rises 3 for every 2 to the right.
• Consider the straight line that passes through A (−1, 0), B (1, 2), and C (4, 5). Plot the points and find the slope of the line.Answer: Slope = CPBP = 5 − 24 − 1 = 1.
• Consider the line in the figure which passes through A (−4, 5) and B (−1, 2). Find the slope of the line.Answer: Slope = 2 − 5−1 − (−4) = −33 = −1, whichever point is taken first.
• But why does ‘m’ represent the slope?Answer: For any two points (x₁, y₁), (x₂, y₂) on y = mx + d: y₂ − y₁x₂ − x₁ = (mx₂ + d) − (mx₁ + d)x₂ − x₁ = m(x₂ − x₁)x₂ − x₁ = m. So the slope is always m, whichever two points we use.
• Consider the line 5x + y = 3. Rewrite it as y = −5x + 3. Can you now find its slope? What does it mean?Answer: Slope m = −5: for every increase of 1 unit in x, y decreases by 5 units. The y-intercept d = 3, so the line crosses the y-axis at (0, 3).
• Can you use the points A, B and C to verify that the slope of this line is indeed −5?Answer: Yes. AB: −2 − 31 − 0 = −5; CA: 3 − 80 − (−1) = −5; CB: −2 − 81 − (−1) = −102 = −5. All three give −5.
• The following diagrams represent ski hills. Rank the hills in order of their steepness, from least to greatest.Answer: Slopes: A = 6070 ≈ 0.86, B = 60110 ≈ 0.55, C = 80100 = 0.8. From least to greatest steepness: B, C, A.
• The ramp at a loading dock rises 2.5 metres over a run of 4 metres. Find the slope of the ramp.Answer: Slope = 2.54 = 58 = 0.625.
• Find the slope of the line l in each of the following diagrams.Answer: (i) 0 (horizontal) (ii) −1 (iii) 23 (iv) undefined (vertical).
• An accessibility ramp for wheelchairs is to be made alongside the staircase. The guidelines given for the slope of the ramp is 1 cm vertical rise to 12 cm horizontal length. What should be the horizontal length of the ramp if the total height of the stairs is 18 cm?Answer: Slope = 112, rise = 18 cm ⇒ horizontal length = 18 × 12 = 216 cm (2.16 m).
• The cost of one paratha and 2 bowls of dahi is ₹90. Also, the cost of 2 parathas and 7 bowls of dahi is ₹260. Can you figure out the cost of one paratha and one bowl of dahi and explain your reasoning?Answer: One bowl of dahi = ₹803 ≈ ₹26.67; one paratha = ₹1103 ≈ ₹36.67. (Doubling the first order gives 2 parathas + 4 dahi = ₹180; the second order has 3 more dahi and costs ₹80 more, so 3 dahi = ₹80.)
• Rakesh’s puzzle: “I have thought of two numbers”, he said. “Their sum is 25 and their difference is 11”.
  We can also solve this using a pair of linear equations.Answer: x + y = 25 and x − y = 11 give x = 18, y = 7. The numbers are 18 and 7.
• Consider another situation. A science exhibition charges an entry fee that is different for adults and children. Here are the amount paid by two different groups.
  Group A: 2 adult tickets and 3 child tickets for ₹600
  Group B: 3 adult tickets and 2 child tickets for ₹700
  How much do individual adult and child tickets cost?
  If the cost of one adult ticket is ₹x and one child ticket is ₹y, then: 2x + 3y = 600, 3x + 2y = 700.
  This gives us another pair of linear equations in two variables. A solution to these equations will give us the answer that we seek. Can you see why?Answer: Adult ticket ₹180, child ticket ₹80. The solution works because only the true prices make both groups’ bills come out right, and those are exactly the pairs that satisfy both equations.
• Given below are some everyday situations. Construct a pair of linear equations in two variables for each situation:
  On two different days a family buys movie tickets and snack boxes. Let the cost of one movie ticket be ₹x and the cost of one snack box be ₹y. Frame a pair of linear equations in x and y to represent the following situations:Answer: (i) 2x + 3y = 850 (ii) 4x + y = 1100.
• Two friends, Sahil and Meena, travelled by taxi. Let the fixed charge for one taxi trip be ₹x and the additional charge per kilometre be ₹y. Frame a pair of linear equations in x and y to represent the following situation:Answer: (i) x + 6y = 122 (ii) x + 8y = 160.
• In a sports meet, tickets for adults cost ₹150 each and tickets for children cost ₹100 each. A total of 200 people attended the meet, and the total amount collected from ticket sales was ₹25,000. (Let the number of adult tickets sold be x and the number of children’s tickets sold be y.) Frame a pair of linear equations in x and y to represent this situation.Answer: x + y = 200 (people) and 150x + 100y = 25000 (money), i.e. 3x + 2y = 500.
• Solve the puzzle and discuss your strategy with your friends. Can you determine the value represented by each shape?Answer: Circle = 4, triangle = 6, rectangle = 8, hexagon = 7; the missing column total is 21.
• Solve the following pair of equations by substitution method:
  7x − 15y = 2 … (1)
  x + 2y = 3 … (2)Answer: x = 4929, y = 1929.
• What if we had expressed y in terms of x, i.e., take Equation (2) and write y = 12(3 − x). Would we still get the same solution?Answer: Yes. Substituting y = 3 − x2 into 7x − 15y = 2 gives 29x = 49, so x = 4929 and y = 1929, the same solution.
• If two angles of a triangle are x° and y° with y = 4x, and the third angle is 50°, what are the angles?Answer: x = 26°, y = 104°; the angles are 26°, 104° and 50°.
• The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save ₹2000 per month, find their monthly incomes.Answer: Monthly incomes: ₹18,000 and ₹14,000.
• The method used in solving this problem is called the Elimination Method because we eliminate one of the variables to obtain a linear equation in the other variable. In the example above, we eliminated y. Try the same problem by eliminating x instead of y. Also try to solve the two equations using the Substitution Method. Assess the pros and cons of each method.Answer: Eliminating x (multiply by 7 and 9) and substitution both give x = 2000, y = 4000, so the incomes are ₹18,000 and ₹14,000 again. Elimination avoids fractions here; substitution is best when a variable has coefficient 1.
• The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?Answer: Taking the tens digit as the larger digit: 42. If the tens digit may be the smaller one, 24 also works. So there are two such numbers, 42 and 24 (one for each choice of which digit is larger).
• Does a pair of linear equations always have a unique solution? Can you analyse this using the Elimination Method? While subtracting the equations to eliminate a particular variable, if the other variable remains, then clearly there is a unique solution. But what if the second variable gets eliminated too while eliminating the first variable? Can you think of a situation where this happens?Answer: No. If one equation’s x– and y-coefficients are the same multiple of the other’s (e.g. x − y = 10 and 10x − 10y = 100), eliminating x removes y too. Then we get either 0 = 0 (infinitely many solutions) or something false like 0 = 1 (no solution).
• How many solutions are there to the equations 10x − 5y = 20 and 4x − 2y = 8?
  Give 3 more examples of pairs of equations that lead to 0 = 0 after subtraction, and therefore have infinitely many solutions. Can you find a simple rule to check when this happens?Answer: Infinitely many: both equations are multiples of 2x − y = 4. Examples: x + y = 3 & 2x + 2y = 6; 3x − y = 1 & 9x − 3y = 3; x − 4y = 2 & −2x + 8y = −4. Rule: a₁a₂ = b₁b₂ = c₁c₂.
• What if one of a₂, b₂, c₂ is zero?Answer: Then the ratio form cannot be used (we cannot divide by 0), but the idea still works in the form a₁ = ka₂, b₁ = kb₂, c₁ = kc₂ (k ≠ 0). So if c₂ = 0, there are infinitely many solutions only if c₁ = 0 too and the other coefficients are proportional; similarly for a₂ or b₂.
• Does a pair of linear equations always have either a unique solution or infinitely many solutions?Answer: No. A pair can have no solution. Example: x − y = 10 and 10x − 10y = 101; 10 × the first gives 10x − 10y = 100, which cannot also equal 101.
• Give 3 more examples of pairs of equations that have no solution. Can you find a simple rule to check when this will happen?Answer: Examples: x + y = 2 & x + y = 5; 2x − 3y = 1 & 4x − 6y = 7; x + 2y = 3 & 3x + 6y = 1. Rule: a₁a₂ = b₁b₂ ≠ c₁c₂.
• Are the converses of the above statements true?Answer: Yes. Unique solution ⇒ a₁a₂ ≠ b₁b₂; infinitely many ⇒ a₁a₂ = b₁b₂ = c₁c₂; no solution ⇒ a₁a₂ = b₁b₂ ≠ c₁c₂. The three conditions cover every pair and never overlap, so each outcome can come from only its own condition.
• Can there be methods other than elimination and substitution to reduce a pair of linear equations in two variables to a linear equation in one variable? If so, describe them.Answer: Yes. For example, the comparison method: make the same variable the subject of both equations and set the two expressions equal. For pairs with swapped coefficients, adding and subtracting the equations gives x + y and x − y directly. (Graphs also give the solution, but they do not reduce the pair to one equation.)
• Consider the following pairs of linear equations in two variables.
  Pair 1: 2x + y = 6 and x − y = 2
  Pair 2: x + y = 4 and 2x + 2y = 8
  Pair 3: 3x − 2y = 6 and 6x − 4y = 12
  Pair 4: x − 2y = 4 and 2x − 4y = 6
  For each pair, prepare a table of values (with at least two ordered pairs). Plot the points on the Cartesian plane. Draw the straight lines.Answer: Pair 1: the lines intersect at (83, 23). Pairs 2 and 3: each pair is the same line (coincident). Pair 4: the lines are parallel.
• How do we find solutions from the graph? What can we say about the number of solutions when the two lines (i) intersect at a point, (ii) are parallel, (iii) are coincident?Answer: A solution is a point lying on both lines. (i) Intersecting at a point: one (unique) solution. (ii) Parallel: no solution. (iii) Coincident: infinitely many solutions.
• Solve the pair of linear equations x + 3y = 6 and 2x − 3y = 12 graphically. Comment on the nature of the solution.Answer: The lines meet at B(6, 0), so x = 6, y = 0; the pair has a unique solution (a₁a₂ = 12 ≠ b₁b₂ = −1).
• Determine if the lines x + 2y − 4 = 0 and 2x + 4y − 12 = 0 intersect, coincide or are parallel to each other. Then verify your answer by graphing the equations.Answer: a₁a₂ = 12, b₁b₂ = 24 = 12, c₁c₂ = −4−12 = 13. Since a₁a₂ = b₁b₂ ≠ c₁c₂, the lines are parallel (no solution). The graph confirms it.
• Can you prove that two lines of equal slope are parallel?
  (Hint: Consider the equations of the two lines to be y = mx + d₁ and y = mx + d₂ )Answer: Take y = mx + d₁ and y = mx + d₂ with d₁ ≠ d₂. A common point would need mx + d₁ = mx + d₂, i.e. d₁ = d₂, which is false. So the lines never meet: they are parallel. (If d₁ = d₂ they are the same line.)
• Romila went to a stationery shop and purchased 2 erasers and 3 A4 sheets for ₹9. Her friend Sonali saw the new variety of erasers and A4 sheets Romila bought, and she also bought 4 erasers and 6 A4 sheets of the same kind for ₹18. Represent this situation graphically.Answer: With eraser ₹x and sheet ₹y: 2x + 3y = 9 and 4x + 6y = 18. Both give the same line (through (0, 3), (3, 1), (4.5, 0)): the lines coincide, so there are infinitely many solutions.
• Form a pair of linear equations for each of the following problems and find their solutions.Answer: (i) −8 and 13 (ii) 39 and 13 (iii) bat ₹1200, ball ₹80 (iv) fixed ₹25, ₹13 per km; 25 km costs ₹350 (v) 79 (vi) 35 (vii) Nuri 50 years, Sonu 20 years (viii) 18 (ix) 10 notes of ₹50 and 15 notes of ₹100 (x) fixed charge ₹15, ₹3 per extra day.
• Form a pair of linear equations and find their common solutions graphically.
  10 students took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.Answer: x + y = 10 and x − y = 4 (x = girls, y = boys). The lines meet at (7, 3): 7 girls and 3 boys.
• Use the ratios a₁a₂, b₁b₂, c₁c₂, to determine whether the lines representing the following pairs of linear equations intersect at a point, are parallel or are coincident.Answer: (i) intersect at a point (ii) coincident (iii) parallel.
• Which of the following pairs of linear equations have solutions? If they have solutions, find them graphically.Answer: (i) Yes, infinitely many (same line, e.g. (0, 5), (2, 3), (5, 0)). (ii) No solution (parallel). (iii) Yes, unique: (2, 2). (iv) No solution (parallel).
• Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.Answer: Length 20 m, width 16 m.
• Given the linear equation 2x + 3y − 8 = 0, write another linear equation in two variables so that the graphs of the pair formed represent (i) intersecting lines (ii) parallel lines (iii) coincident lines.Answer: (i) 3x − 2y − 7 = 0 (intersecting) (ii) 4x + 6y − 12 = 0 (parallel) (iii) 4x + 6y − 16 = 0 (coincident). Many other answers are possible.
• Here is a problem that was posed by Mahāvīrāchārya in Gaṇita sāra saṅgraha (c. 850 CE).
  The price of 9 citrons and 7 fragrant wood-apples taken together is 107; and the price of 7 citrons and 9 fragrant wood-apples taken together is 101.
  O mathematician, tell me quickly the price of each citron and of each fragrant wood-apple.
  (Hint: The given problem can be modelled as
  9x + 7y = 107
  7x + 9y = 101
  Can you figure out a way of solving these equations without directly using elimination or the substitution method? What is special in this pair of equations? The x and y coefficients are interchanged in the 2 equations.
  What will happen if we add the pair of equations? What will happen if we subtract the pair of equations? Can the resulting equations be solved to find the values of x and y?)Answer: Adding: 16(x + y) = 208 ⇒ x + y = 13. Subtracting: 2(x − y) = 6 ⇒ x − y = 3. So a citron costs 8 and a wood-apple 5.
• 5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of each pencil and pen.Answer: Pencil ₹3, pen ₹5.
• Find the height of the stool.Answer: Stool + cat = 85 and stool − cat = 25, so the stool is 55 cm high (and the cat 30 cm).
• The graph of the line y = 3x, passing through (0, 0) and (2, 6) is given below.Answer: (i) Slope = 6 − 02 − 0 = 3 (ii) y-intercept 0: the line passes through the origin (iii) (a) 15 cm (b) 7 minutes (iv) No: 3 × 4 = 12 ≠ 10 (v) (3, 9).
• In countries like the USA, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Celsius to Fahrenheit:
  F = 95C + 32Answer: (i) A straight line through (0, 32) and (−40, −40), slope 95 (ii) 86 °F (iii) 35 °C (iv) 0 °C = 32 °F; 0 °F = −1609 ≈ −17.8 °C (v) Yes, −40 (−40 °C = −40 °F).
• Solve the following system of equations graphically:
  2x + y = 6, 2x − y − 2 = 0.Answer: The lines meet at (2, 2): x = 2, y = 2.
• Find the point of intersection of the lines shown on the cover page.Answer: The lines cross at (9, 2), which satisfies 15x − 10y − 115 = 0. This is read from the graph and checked in the labelled equation.
• Give a formula to find the x-intercept of the line y = mx + c.Answer: Put y = 0: 0 = mx + c ⇒ x = −cm (m ≠ 0). The line crosses the x-axis at (−cm, 0).
• A person is choosing between two mobile plans.
  Plan A: ₹50 monthly fee + ₹0.20 per minute of call time.
  Plan B: ₹30 monthly fee + ₹0.30 per minute of call time.
  For how many minutes of calling per month is Plan A cheaper than Plan B? For how many minutes is Plan B cheaper? Also find the number of minutes at which both plans cost the same.Answer: Both cost ₹90 at 200 minutes. Plan A is cheaper for more than 200 minutes; Plan B is cheaper for fewer than 200 minutes.
• How many lines exist that
  (i) have a given slope?
  (ii) have a given slope and pass through a given point?Answer: (i) Infinitely many (all parallel to each other). (ii) Exactly one.
• For what values of p does the pair of equations given below have a unique solution?
  4x + py + 8 = 0; 2x + 2y + 2 = 0.Answer: Unique solution when 42 ≠ p2, i.e. for all p ≠ 4.
• Find the values of a and b for which the following system of equations has infinitely many solutions:
  (a + b)x − 2by = 5a + 2b + 1; 3x − y = 14.Answer: a = 5, b = 1 (the first equation becomes 6x − 2y = 28, which is 2 × (3x − y = 14)).
• Find the value of ‘k’ for which the following system of equations represents a pair of coincident lines:
  x + 2y = 3; (k − 1)x + (k + 1)y = k + 3.Answer: k = 3 (the second equation becomes 2x + 4y = 6).
• (i) Robot 1 starts from the origin and traces a path by repeatedly moving 3 units to the right and then 4 units upward. Robot 2 starts from the point (10, 0) and repeatedly moves 1 unit to the right and then 2 units upward. Will the paths traced by these two robots intersect and if so, where?
  (ii) Robot 1 starts from (3, 0) and traces a path by repeatedly moving 5 units to the right and then 5 units upward. Robot 2 starts from the point (7, 0) and repeatedly moves 5 units to the right and then 3 units downwards. Will the paths traced by these two robots intersect and if so, where?Answer: (i) Yes, at (30, 40): Robot 1 follows y = 43x and Robot 2 follows y = 2(x − 10). (ii) No, as straight tracks: the lines y = x − 3 and y = −35(x − 7) would meet only at (4.5, 1.5), behind Robot 2’s start, where Robot 2 never goes. (Traced step by step, Robot 1’s first move along the x-axis, from (3, 0) to (8, 0), passes over Robot 2’s starting point (7, 0), so the two step paths share only that short stretch from (7, 0) to (8, 0).)
• At a certain time, Jacob notices that his digital watch reads ‘a’ minutes after two o’clock. Fifteen minutes later, it reads ‘b’ minutes after three o’ clock. He noticed that a is six times greater than b. What time was it when he looked at his watch for the second time?Answer: a − b = 45 and a = 6b give b = 9, a = 54. The second time was 9 minutes past 3 (3:09).
• The sum of the digits of a two-digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. Find the number.Answer: 78 (its reverse 87 is 9 more, and 7 + 8 = 15).
• In a cyclic quadrilateral ABCD, ∠A = (x + 7)°, ∠B = (y + 8)°, ∠C = (3y + 23)° and ∠D = (4x + 12)°. Find all four angles of the cyclic quadrilateral.Answer: x = 30, y = 40: ∠A = 37°, ∠B = 48°, ∠C = 143°, ∠D = 132°.
• A train moving with uniform speed for a certain distance takes 6 hours less if its speed is increased by 6 km/hour. It would have taken 6 hours more had its speed been decreased by 4 km/hour. Find the distance travelled and the speed of the train.Answer: Speed 24 km/h, time 30 h, distance 720 km.
• The age of a father is equal to the sum of the ages of his four children. After 20 years, the sum of the ages of the children will be twice the age of the father. Find the age of the father.Answer: The father is 40 years old.

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