Pair of Linear Equations in Two Variables · Class 10 Mathematics: 220 MCQs with answers

Q1. A linear equation in two variables has the form

Q2. A solution of a pair of linear equations is a pair (x, y) that

Q3. The graph of a linear equation in two variables is

Q4. If two lines cross at exactly one point, the pair of equations has

Q5. If the two lines are parallel, the pair has

Q6. If the two lines coincide, the number of solutions is

Q7. The point where the line $x + 3y = 6$ meets the y-axis is

Q8. The point where $2x - 3y = 12$ meets the x-axis is

Q9. A pair of equations that has at least one solution is called

Q10. A pair with no solution is called

Q11. Which point lies on the line $2x + 3y = 12$?

Q12. Assertion (A): One linear equation in two variables has infinitely many solutions. Reason (R): Every point on its graph, which is a straight line, satisfies the equation.

Q13. How many solutions does the single equation $x + y = 7$ have?

Q14. The line $y = 4$ is

Q15. Which equation is satisfied by the point $(0, 0)$?

Q16. Form a pair of linear equations: the cost of 2 kg of wheat and 3 kg of rice is ₹140, and the cost of 4 kg of wheat and 5 kg of rice is ₹260.

Q17. Find two solutions of the equation $2x + y = 8$ and use them to describe its line.

Q18. The sum of two numbers is 20 and their difference is 4. Form a pair of equations and solve it.

Q19. A pair of linear equations has exactly one solution when

Q20. A pair has infinitely many solutions when

Q21. A pair has no solution when

Q22. Coincident lines correspond to

Q23. Parallel lines correspond to a pair that is

Q24. Written in standard form, $3x + 2y = 7$ has c equal to

Q25. For the pair $x + y - 5 = 0$ and $2x + 2y - 10 = 0$, the ratio $\frac{c_1}{c_2}$ is

Q26. A dependent pair of linear equations is

Q27. Which case does NOT need the ratio of the constants?

Q28. Assertion (A): The pair $2x + 3y - 5 = 0$ and $4x + 6y - 10 = 0$ has infinitely many solutions. Reason (R): All three ratios of the coefficients are equal.

Q29. A pair of equations represents two lines that meet at exactly one point. The pair is

Q30. Without drawing, say how many solutions $x - 2y = 0$ and $3x + 4y = 20$ have, and name the pair.

Q31. Classify the pair $2x + 3y - 9 = 0$ and $4x + 6y - 18 = 0$, giving all three ratios.

Q32. Show that $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ is an inconsistent pair.

Q33. State the conditions on the ratios for a pair to be (i) consistent with a unique solution (ii) inconsistent.

Q34. The first step of the substitution method is to

Q35. Rearranging $x + 2y = 3$ for x gives

Q36. Substituting $x = 3 - 2y$ into $7x - 15y = 2$ gives

Q37. $7(3 - 2y)$ equals

Q38. You must substitute the expression into

Q39. Solving $y = 2x$ and $x + y = 9$ by substitution gives

Q40. Which variable is easiest to make the subject in $3x + 4y = 10$ and $x - 2y = 1$?

Q41. After substituting, the variables cancel and you are left with $5 = 5$. The pair has

Q42. After substituting, you are left with $4 = 9$. The pair has

Q43. Where should the final answer be checked?

Q44. Assertion (A): Substituting $x = 3 - 2y$ back into $x + 2y = 3$ is useless. Reason (R): It gives the identity $3 = 3$, which contains no new information.

Q45. Making y the subject of $5x - y = 12$ gives

Q46. Which line of working shows Step 2 of the method?

Q47. Solve by substitution: $x + y = 14$ and $x - y = 4$.

Q48. The sum of two numbers is 45 and one is four times the other. Find them by substitution.

Q49. The first step of the elimination method is to

Q50. If the matched terms are $+5y$ and $+5y$, you should

Q51. If the matched terms are $+4x$ and $-4x$, you should

Q52. Multiplying $2x + 3y = 13$ by 3 gives

Q53. Adding $x + y = 7$ and $x - y = 3$ gives

Q54. Solving $x + y = 7$ and $x - y = 3$ gives

Q55. To eliminate y from $2x + 3y = 13$ and $5x - 2y = 4$, the multipliers are

Q56. Reversing the digits of the number $10x + y$ produces

Q57. Elimination leaves $0 = 0$. The pair has

Q58. Elimination leaves $0 = 5$. The pair has

Q59. Assertion (A): An equation may be multiplied throughout by 3 during elimination. Reason (R): Multiplying every term by a non-zero number gives an equivalent equation with the same solutions.

Q60. Subtracting $x - y = 2$ from $x + y = 8$ gives

Q61. Which is NOT a step of the elimination method?

Q62. Two pencils and three erasers cost ₹9. With ₹x per pencil and ₹y per eraser, the equation is

Q63. 'The number of skirts is two less than twice the number of pants' becomes

Q64. The lines $y = 2x - 2$ and $y = 4x - 4$ meet at

Q65. The solution of $x + 3y = 6$ and $2x - 3y = 12$ is

Q66. Which pair of points can be used to draw $3x + 4y = 12$?

Q67. A father is 30 years older than his daughter. With ages x and y, the equation is

Q68. If $(2, 3)$ is a solution of $ax + 3y = 13$, then a is

Q69. The pair $2x + 3y = 9$ and $4x + 6y = 18$ represents

Q70. Which of these pairs is inconsistent?

Q71. The solution of $x + y = 5$ and $x - y = 1$ is

Q72. A line passes through $(0, 5)$ and $(5, 0)$. Its equation is

Q73. If a pair of lines coincides, the pair of equations is

Q74. Which situation needs a PAIR of equations?

Q75. Assertion (A): The pair $x + y = 5$ and $2x + 2y = 10$ has exactly one solution. Reason (R): The second equation is twice the first, so both represent the same line.

Q76. Assertion (A): The point $(2, 3)$ is a solution of the pair $x + y = 5$ and $x - y = 1$. Reason (R): A solution of a pair must satisfy both equations.

Q77. The equations $x + y = 6$ and $x + y = 9$ represent

Q78. A pair of equations is formed from two facts about a situation. If the second fact repeats the first, the pair will have

Q79. A two-digit number has tens digit x and units digit y. The number is

Q80. Solve graphically (describing the two lines): $x + y = 7$ and $x - y = 1$.

Q81. Champa buys two fewer skirts than twice the number of pairs of trousers, and also four fewer than four times that number. Form the equations and solve them.

Q82. Show that the pair $2x + 3y = 9$ and $4x + 6y = 18$ has infinitely many solutions.

Q83. Show that the pair $x + 2y = 4$ and $2x + 4y = 12$ has no solution.

Q84. Explain why the graphical method is unsuitable for $7x - 15y = 2$ and $x + 2y = 3$.

Q85. (a) Write the general form of a pair of linear equations in two variables and say what a solution of the pair means. (b) Solve graphically (describing the lines): $x + 3y = 6$ and $2x - 3y = 12$. (c) Verify the solution in both equations and state whether the pair is consistent.

Q86. A shopkeeper sells 2 pens and 3 notebooks for ₹80, and 3 pens and 5 notebooks for ₹130. (a) Form a pair of linear equations, stating clearly what your letters mean. (b) Solve the pair. (c) Find the cost of 1 pen and 2 notebooks together.

Q87. The pair $x - 2y = 0$ and $3x + 4y - 20 = 0$ has

Q88. The pair $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ has

Q89. For $3x + 2y = 5$ and $6x + 4y = 11$, the number of solutions is

Q90. For $2x + 3y - 8 = 0$ and $4x + 6y - 16 = 0$, the lines are

Q91. For which value of k does $2x + ky = 5$ and $4x + 6y = 11$ have a unique solution?

Q92. For which k does $x + 2y = 3$ and $5x + ky + 7 = 0$ have no solution?

Q93. If a pair of equations is consistent with a unique solution, then

Q94. The pair $5x - 8y + 1 = 0$ and $3x - \frac{24}{5}y + \frac{3}{5} = 0$ has

Q95. Two lines have the same slope but different intercepts. The pair is

Q96. For $9x - 4y = 2000$ and $7x - 3y = 2000$, the number of solutions is

Q97. Which pair represents intersecting lines?

Q98. Assertion (A): The pair $x + y = 3$ and $2x + 2y = 7$ is inconsistent. Reason (R): All three of the coefficient ratios are equal.

Q99. Assertion (A): If $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, the pair has infinitely many solutions. Reason (R): Equal ratios of a and b mean the lines have the same slope.

Q100. A pair of equations that is consistent may have

Q101. Which pair of lines is coincident?

Q102. If both equations of a pair are multiplied by 3, the number of solutions

Q103. For $x + y = 5$ and $3x + 3y = 15$, the value of $\frac{c_1}{c_2}$ is

Q104. Assertion (A): For $4x + 6y = 12$ and $2x + 3y = 6$ the lines coincide. Reason (R): Dividing the first equation by 2 gives the second.

Q105. Find the value of k for which $2x + ky = 5$ and $4x + 6y = 11$ have no solution.

Q106. The pair $3x + 2y = 5$ and $6x + 4y = 11$ is given. Decide the number of solutions and explain what the graph looks like.

Q107. Why is it not enough to check only $\frac{a_1}{a_2} = \frac{b_1}{b_2}$ before answering 'infinitely many solutions'?

Q108. Find a value of c for which $2x + 3y = 7$ and $4x + 6y = c$ has (i) infinitely many solutions (ii) no solution.

Q109. (a) State the three conditions on the ratios of coefficients, with the picture and the number of solutions in each case. (b) Apply them to: (i) $x - 2y = 0$, $3x + 4y = 20$ (ii) $2x + 3y = 9$, $4x + 6y = 18$ (iii) $x + 2y = 4$, $2x + 4y = 12$.

Q110. Two friends compare mobile plans. Plan A costs ₹x fixed plus ₹y per GB and Plan B is described by the equation $2x + 6y = 700$; Plan A's bill for 3 GB is given by $x + 3y = 350$. (a) Write both equations in standard form and find the three ratios. (b) How many solutions does the pair have, and what does that mean about the two descriptions? (c) Give one change to the second equation that would make the pair inconsistent, and say what that would mean.

Q111. Solving $x + y = 14$ and $x - y = 4$ by substitution gives

Q112. The solution of $s - t = 3$ and $\frac{s}{3} + \frac{t}{2} = 6$ is

Q113. Solving $0.2x + 0.3y = 1.3$ and $0.4x + 0.5y = 2.3$ gives

Q114. For $2x + 3y = 9$ and $4x + 6y = 18$, substitution ends with

Q115. For $2x + 3y = 9$ and $4x + 6y = 20$, substitution ends with

Q116. Aftab's ages give $x - 7y = -42$ and $x - 3y = 6$. His present age is

Q117. A two-digit number has digits adding to 9, and it stands 27 above the number its digits make in reverse. The number is

Q118. If $x = 2y$ and $3x - y = 10$, then y is

Q119. A number is three times another and their sum is 48. The smaller number is

Q120. Which pair is best suited to substitution?

Q121. Solving $y = 3x - 2$ and $2x + y = 8$ gives

Q122. Assertion (A): A fractional answer such as $x = \frac{49}{29}$ means a mistake has been made. Reason (R): The solution of a pair of linear equations is always a whole number.

Q123. Assertion (A): In the substitution method, the expression must be put into the equation that was NOT rearranged. Reason (R): Substituting into the rearranged equation gives an identity.

Q124. Solving $x - y = 1$ and $2x + y = 8$ by substitution gives

Q125. For $x + y = 6$ and $2x + 2y = k$, the pair has NO solution when k is

Q126. Five pens and three pencils cost ₹44, and one pen costs ₹4 more than one pencil. The cost of a pen is

Q127. The sum of two numbers is 35 and their difference is 13. The larger number is

Q128. Solve by substitution: $3x - y = 3$ and $9x - 3y = 9$.

Q129. Solve by substitution: $x + 2y = 5$ and $2x + 4y = 15$.

Q130. Solve by substitution: $7x - 15y = 2$ and $x + 2y = 3$.

Q131. Solve by substitution: $s - t = 3$ and $\frac{s}{3} + \frac{t}{2} = 6$.

Q132. The larger of two supplementary angles exceeds the smaller by 18°. Find both angles by substitution.

Q133. Explain what it means if, while solving by substitution, both variables disappear and you are left with $0 = 0$.

Q134. (a) Write the three steps of the substitution method. (b) Use it to solve $2x + 3y = 11$ and $2x - 4y = -24$. (c) Verify your answer in both original equations.

Q135. (a) Solve by substitution: $3x - y = 3$, $9x - 3y = 9$, and say what your result means. (b) Solve by substitution: $x + 2y = 5$, $2x + 4y = 15$, and say what your result means. (c) What picture do the lines make in each case?

Q136. Solving $x + y = 5$ and $2x - 3y = 4$ by elimination gives

Q137. Solving $3x + 4y = 10$ and $2x - 2y = 2$ gives

Q138. Solving $3x - 5y = 4$ and $9x = 2y + 7$ gives

Q139. Two people's incomes are in the ratio 9 : 7 and their spendings in the ratio 4 : 3, each saving ₹2000 a month. Their incomes are

Q140. Adding a two-digit number to the number its digits make in reverse gives 66, and the two digits are 2 apart. One such number is

Q141. Five years ago a father was seven times as old as his son; in five years he will be three times as old. The son's present age is

Q142. Which pair is better suited to elimination than to substitution?

Q143. Half the perimeter of a rectangular garden is 36 m and its length is 4 m more than its width. Its dimensions are

Q144. Subtracting $(5x + 3y = 13)$ from $(5x - 2y = 4)$ gives

Q145. Two kilograms of apples and one of grapes cost ₹160, while four kilograms of apples and two of grapes cost ₹300. The pair has

Q146. Assertion (A): In elimination you add the equations when the matched coefficients have the same sign. Reason (R): Adding two equal positive terms gives zero.

Q147. Assertion (A): Elimination and substitution give the same solution for any pair. Reason (R): Both methods only ever replace an equation by an equivalent one.

Q148. For what value of k do the lines $x + 2y = 3$ and $2x + ky = 6$ coincide?

Q149. The lines $x = 3$ and $y = -2$ meet at

Q150. A pair of equations has the solution $(4, 1)$. Which pair could it be?

Q151. The cost of 3 kg apples and 2 kg grapes is ₹180, and of 2 kg apples and 4 kg grapes is ₹200. Taking prices x and y per kg, the pair is

Q152. Two lines meet at $(2, 5)$. How many pairs of linear equations have this solution?

Q153. If a pair of lines has exactly one common point, which is NOT possible?

Q154. Assertion (A): Two distinct straight lines can meet at exactly two points. Reason (R): Two points determine a unique straight line.

Q155. A boat's speed downstream and upstream gives the pair $x + y = 10$ and $x - y = 6$, where x is the boat's speed and y the stream's. The stream's speed is

Q156. Which pair of equations is NOT a pair of linear equations in two variables?

Q157. If $(3, 2)$ is a solution of $2x + ky = 12$, find k and write one more solution of that equation.

Q158. A father is three times as old as his son. Five years later he will be two and a half times as old. Form the pair of equations (do not solve).

Q159. (a) Describe the three ways in which the lines of a pair of linear equations can be placed, and give the number of solutions in each case. (b) For each case, give one example of a pair of equations. (c) Which of your three pairs are consistent?

Q160. The coach of a cricket team buys 3 bats and 6 balls for ₹3900. Later she buys another bat and 3 more balls of the same kind for ₹1300. (a) Represent this situation algebraically. (b) Find the cost of one bat and one ball by any method. (c) If she now buys 2 bats and 5 balls, what will she pay?

Q161. For what value of k do $2x + 3y - 7 = 0$ and $(k-1)x + (k+2)y - 3k = 0$ have infinitely many solutions?

Q162. For which value of k does $kx + 2y = 5$ and $3x + y = 1$ have a unique solution?

Q163. The pair $2x + 3y = 7$ and $(a - b)x + (a + b)y = 3a + b - 2$ has infinitely many solutions. Then

Q164. If the lines $3x + 2ky = 2$ and $2x + 5y + 1 = 0$ are parallel, then k is

Q165. A pair with $a_1b_2 - a_2b_1 = 0$ and $b_1c_2 - b_2c_1 \neq 0$ has

Q166. Assertion (A): A pair of linear equations can have exactly two solutions. Reason (R): Two straight lines can cross at no point, one point, or infinitely many.

Q167. For $x + 2y = 3$ and $2x + 4y = k$, the pair has infinitely many solutions when k is

Q168. Which change to $x + y = 4$ and $2x + 2y = 8$ would make the pair inconsistent?

Q169. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{1}{2}$, then the second equation is

Q170. Two equations with $a_1 = 2$, $b_1 = -3$, $a_2 = -4$, $b_2 = 6$ represent lines that are

Q171. Which of these pairs is consistent?

Q172. For what value of k does $kx + 3y = k - 3$ and $12x + ky = k$ have infinitely many solutions?

Q173. Two lines are given by $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ with $a_2 = 0$. Explain how to test for a unique solution without dividing by zero.

Q174. (a) Find the value of k for which the pair $3x + y = 1$ and $(2k - 1)x + (k - 1)y = 2k + 1$ has no solution. (b) Show that for this k the lines are parallel and not coincident. (c) For what values of k does the pair have a unique solution?

Q175. (a) For what value of k does $x + 2y = 3$ and $5x + ky + 7 = 0$ have no solution? (b) For that k, write both equations in standard form and give all three ratios. (c) Is there any value of k for which this pair has infinitely many solutions? Justify your answer.

Q176. If $2x + 3y = 11$ and $2x - 4y = -24$, then the value of $y$ is

Q177. For the pair $x + y = 5$ and $2x + 2y = k$, substitution leaves a true statement when k is

Q178. If (x, y) is the solution of $3x + 2y = 11$ and $2x + 3y = 4$, then $x - y$ equals

Q179. A fraction becomes $\frac{4}{5}$ if 1 is added to both numerator and denominator, and $\frac{1}{2}$ if 5 is subtracted from both. The fraction is

Q180. Two numbers differ by 26 and one is three times the other. The larger number is

Q181. The larger of two supplementary angles exceeds the smaller by 18°. The larger angle is

Q182. Assertion (A): The pair $y = 2x + 1$ and $y = 2x + 5$ has no solution. Reason (R): Substituting gives $2x + 1 = 2x + 5$, i.e. $1 = 5$, which is false.

Q183. For which value of m does $y = mx + 3$ and $y = 2x + 3$ have infinitely many solutions?

Q184. The cost of 2 kg of apples and 1 kg of grapes is ₹160, and of 4 kg of apples and 2 kg of grapes is ₹300. The pair has

Q185. Assertion (A): Substitution works on any pair of linear equations in two variables. Reason (R): One variable can always be written in terms of the other when its coefficient is not zero.

Q186. In solving $3x + y = 10$ and $x + y = 6$ by substitution, the cleanest first move is

Q187. Aftab is seven times as old as his daughter was seven years ago, and three times as old as she will be in three years. Find their present ages.

Q188. A fraction becomes $\frac{4}{5}$ when 1 is added to both its numerator and denominator, and $\frac{1}{2}$ when 5 is subtracted from both. Find the fraction.

Q189. A two-digit number is such that the sum of its digits is 9, and the number is 27 more than the number obtained by reversing its digits. (a) Taking the tens digit as x and the units digit as y, write both equations. (b) Solve them by substitution. (c) State the number and verify both conditions.

Q190. A lending library charges a fixed amount for the first three days and an extra charge for each further day. Saritha paid ₹27 for a book kept for seven days, and Susy paid ₹21 for a book kept for five days. (a) Taking the fixed charge as ₹x and the charge per extra day as ₹y, write the two equations. (b) Solve them by substitution. (c) What would a book kept for ten days cost?

Q191. Twice a number added to three times a second gives 92, while four times the first minus the second gives 16. The first number is

Q192. A boat covers 30 km upstream and 44 km downstream in 10 hours, and 40 km upstream and 55 km downstream in 13 hours. The boat's speed in still water is

Q193. For what value of k do $2x + 3y = 7$ and $4x + 6y = k$ have infinitely many solutions?

Q194. Eliminating x from $ax + by = c$ and $a'x + b'y = c'$ requires multiplying by

Q195. Two complementary angles differ by 20°. Eliminating gives the larger angle as

Q196. Five books and seven pens together cost ₹79, while seven books and five pens cost ₹77. The cost of one book is

Q197. A two-digit number is four times the sum of its digits, and twice the product of its digits. The number is

Q198. Assertion (A): For $3x + 2y = 8$ and $6x + 4y = 16$, elimination leaves $0 = 0$. Reason (R): The two equations represent parallel lines that never meet.

Q199. In a 3-mark word problem, the mark most often lost is for

Q200. The coach of a team buys 7 bats and 6 balls for ₹3800, and later 3 bats and 5 balls for ₹1750. The cost of one bat is