1-Mark Questions (20) — with Answers
Q1. Is \(2x+3y=5\) a linear equation in two variables?
Ans. Yes. Powers of \(x, y\) are 1 (general form \(ax+by+c=0\)).
Q2. Does \((3,4)\) satisfy \(x+y=7\)?
Ans. \(3+4=7\Rightarrow\) Yes, it satisfies.
Q3. Check: Does \((2,1)\) satisfy \(2x-3y=1\)?
Ans. \(2(2)-3(1)=4-3=1\Rightarrow\) Yes.
Q4. If \(x+y=14\) and \(x=8\), find \(y\).
Ans. \(8+y=14\Rightarrow y=6\).
Q5. If \(3x+y=15\) and \(y=6\), find \(x\).
Ans. \(3x+6=15\Rightarrow 3x=9\Rightarrow x=3\).
Q6. How many solutions does \(x+y=10\) have?
Ans. Infinitely many ordered pairs \((x,y)\).
Q7. Write general form of a linear equation in two variables.
Ans. \(ax+by+c=0\), where not both \(a,b\) are zero.
Q8. Express “sum of two numbers is 20” as an equation.
Ans. \(x+y=20\).
Q9. If \((10,8)\) satisfies \(x-y=2\), does \((8,10)\) satisfy it?
Ans. No. \(8-10=-2\neq 2\).
Q10. Check: \((1,1)\) for \(8x+3y=11\).
Ans. \(8(1)+3(1)=11\Rightarrow\) Yes.
Q11. Coefficient of \(y\) in \(2x-5y+7=0\)?
Ans. \(-5\).
Q12. Convert \(y=3x-2\) to general form.
Ans. \(3x-y-2=0\).
Q13. Give one solution of \(x+y=5\) with \(x<0\).
Ans. \((-1,6)\) (many others possible).
Q14. Find \(k\) if \((k,2)\) satisfies \(2x+3y=16\).
Ans. \(2k+6=16\Rightarrow k=5\).
Q15. Put \(3x+2y-12=0\) in the form \(y=mx+c\).
Ans. \(2y=12-3x\Rightarrow y=6-\tfrac{3}{2}x\).
Q16. A unique solution of two linear equations corresponds to what graph?
Ans. Two intersecting non-parallel lines (one common point).
Q17. If \(2x+3y=12\) and \(x=0\), find \(y\).
Ans. \(3y=12\Rightarrow y=4\).
Q18. If \(x-y=5\) and \(y=-2\), find \(x\).
Ans. \(x-(-2)=5\Rightarrow x=3\).
Q19. Does \((2,-1)\) satisfy \(3x+y=5\)?
Ans. \(3(2)+(-1)=6-1=5\Rightarrow\) Yes.
Q20. If \(4m-3n=2\) and \(n=2\), find \(m\).
Ans. \(4m-6=2\Rightarrow 4m=8\Rightarrow m=2\).
2-Mark Questions (20) — with Solutions
Q1. Solve \(x+y=14,\; x-y=2\).
Ans. Add: \(2x=16\Rightarrow x=8\). Then \(8+y=14\Rightarrow y=6\). Hence \((8,6)\).
Q2. Solve \(3x+y=5,\; 2x+3y=1\).
Ans. From eqn1: \(y=5-3x\). Substitute: \(2x+3(5-3x)=1\Rightarrow -7x=-14\Rightarrow x=2\), so \(y=-1\). \((2,-1)\).
Q3. Solve \(3x-4y-15=0,\; x+y=-2\).
Ans. \(3x-4y=15\). From second: \(x=-2-y\). Substitute: \(3(-2-y)-4y=15\Rightarrow -6-3y-4y=15\Rightarrow y=-3\), \(x=1\). \((1,-3)\).
Q4. Solve \(8x+3y=11,\; 3x-y=2\).
Ans. From second: \(y=3x-2\). Sub: \(8x+3(3x-2)=11\Rightarrow 17x=17\Rightarrow x=1\). Then \(y=1\). \((1,1)\).
Q5. Solve \(3x-4y=16,\; 2x-3y=10\).
Ans. From first: \(3x=16+4y\Rightarrow x=\tfrac{16+4y}{3}\). Sub: \(2\tfrac{16+4y}{3}-3y=10\Rightarrow 32+8y-9y=30\Rightarrow y=2\), then \(x=8\). \((8,2)\).
Q6. Solve \(2x-7y=7,\; 3x+y=22\).
Ans. From second: \(y=22-3x\). Sub: \(2x-7(22-3x)=7\Rightarrow 23x=161\Rightarrow x=7\), \(y=1\). \((7,1)\).
Q7. Solve \(x-2y=-2,\; x+2y=10\).
Ans. Add: \(2x=8\Rightarrow x=4\). Then \(4-2y=-2\Rightarrow y=3\). \((4,3)\).
Q8. Ages: \(x+y=45\), and \(2x-y=54\). Find \(x,y\).
Ans. Add: \(3x=99\Rightarrow x=33\). Then \(33+y=45\Rightarrow y=12\). Mother 33, son 12.
Q9. Two-digit number: \(10y+x=4(x+y)+3\), and \(10y+x+18=10x+y\). Find number.
Ans. From first: \(x-2y=-1\). From second: \(x-y=2\). Solve ⇒ \(y=3, x=5\). Number \(=10y+x=35\).
Q10. Solve \(2x+y=5\) and \(3x-y=5\).
Ans. Add: \(5x=10\Rightarrow x=2\), \(2(2)+y=5\Rightarrow y=1\). \((2,1)\).
Q11. Solve \(x-2y=-1\) and \(2x-y=7\).
Ans. \(x=2y-1\Rightarrow 2(2y-1)-y=7\Rightarrow 3y=9\Rightarrow y=3, x=5\). \((5,3)\).
Q12. Solve \(x+y=11\) and \(2x-3y=7\).
Ans. \(x=11-y\Rightarrow 22-5y=7\Rightarrow y=3\Rightarrow x=8\). \((8,3)\).
Q13. Solve \(2x+y=-2\) and \(3x-y=7\).
Ans. Add: \(5x=5\Rightarrow x=1\). Then \(2(1)+y=-2\Rightarrow y=-4\). \((1,-4)\).
Q14. Solve \(2x-y=5\) and \(3x+2y=11\).
Ans. From first: \(y=2x-5\). Sub: \(3x+2(2x-5)=11\Rightarrow 7x=21\Rightarrow x=3, y=1\). \((3,1)\).
Q15. Convert: “Sum of length \(x\) and breadth \(y\) is 23 after reducing each by 5, perimeter \(=26\).” Find \(x+y\).
Ans. \(2[(x-5)+(y-5)]=26\Rightarrow x+y=23\).
Q16. Find one solution of \(3x-5y=16\) with \(x=1\).
Ans. \(3(1)-5y=16\Rightarrow -5y=13\Rightarrow y=-\tfrac{13}{5}\). So \((1,-\tfrac{13}{5})\).
Q17. Put \(2x+3y+4=0\) into \(y=mx+c\).
Ans. \(3y=-2x-4\Rightarrow y=-\tfrac{2}{3}x-\tfrac{4}{3}\).
Q18. Verify if \((9,5)\) satisfies \(x+y=14\) and \((10,4)\) also does.
Ans. \(9+5=14\) ✓ and \(10+4=14\) ✓. Both are solutions.
Q19. Check common solution of \(x+y=14\) and \(x-y=2\).
Ans. Unique common solution is \((8,6)\) (satisfies both).
Q20. General form \(ax+by+c=0\). Identify \(a,b,c\) for \(4x-y+7=0\).
Ans. \(a=4,\; b=-1,\; c=7\).
3-Mark Questions (20) — with Solutions
Q1. Find two numbers whose sum is \(36\). If 9 is subtracted from 8 times the first, we get the second.
Ans. Let numbers be \(x,y\). \(x+y=36\), \(y=8x-9\). Add: \(x+8x-9=36\Rightarrow 9x=45\Rightarrow x=5\), so \(y=31\).
Q2. Sum of two numbers is \(103\). Dividing greater by smaller gives quotient \(2\) and remainder \(19\). Find numbers.
Ans. Let greater \(x\), smaller \(y\). \(x+y=103\), and \(x=2y+19\). Sub: \(2y+19+y=103\Rightarrow 3y=84\Rightarrow y=28\), \(x=75\).
Q3. Salil’s age is 23 more than half Sangram’s. Five years ago their sum was 55. Find present ages.
Ans. Let ages be \(x\) (Salil), \(y\) (Sangram). Given \(x=\tfrac{y}{2}+23\Rightarrow 2x-y=46\). Also \((x-5)+(y-5)=55\Rightarrow x+y=65\). Add: \(3x=111\Rightarrow x=37\). Then \(y=28\).
Q4. A two-digit number is 4 times the sum of its digits; after interchanging digits, the new number is 9 less than \(4\) times original. Find number.
Ans. Let tens \(y\), units \(x\). \(10y+x=4(x+y)\Rightarrow x-2y=0\). Also \(10x+y=4(10y+x)-9\Rightarrow 8x-19y=-9\). Solving gives \(y=3, x=6\). Number \(=36\).
Q5. Population of a town was 50,000. In a year, males increased by 5% and females by 3%, total became 52,020. Find last year’s males and females.
Ans. Let males \(x\), females \(y\). \(x+y=50000\). Next year: \(\tfrac{105}{100}x+\tfrac{103}{100}y=52020\Rightarrow 105x+103y=5\,202\,000\). Subtract \(103(x+y)=5\,150\,000\): \(2x=52\,000\Rightarrow x=26\,000\), so \(y=24\,000\).
Q6. In a zoo, lions and peacocks total \(50\); total legs \(140\). Find counts.
Ans. Let lions \(x\) (4 legs), peacocks \(y\) (2 legs). \(x+y=50\), \(4x+2y=140\Rightarrow 2x+y=70\). Subtract: \(x=20\), \(y=30\).
Q7. Envelope has ₹5 and ₹10 notes totalling ₹350. No. of ₹5 notes is \(10\) less than twice the no. of ₹10 notes. Find counts.
Ans. Let ₹5 notes \(x\), ₹10 notes \(y\). \(5x+10y=350\Rightarrow x+2y=70\); and \(x=2y-10\). So \(2y-10+2y=70\Rightarrow y=20\), \(x=30\).
Q8. Denominator is 1 less than twice numerator. If 1 is added to each, the ratio becomes \(\tfrac{3}{5}\). Find fraction.
Ans. \(y=2x-1\); and \(\tfrac{x+1}{y+1}=\tfrac{3}{5}\Rightarrow 5x+5=3y+3\). With \(y=2x-1\): \(5x+5=6x-3+3\Rightarrow x=5\Rightarrow y=9\). Fraction \(=\tfrac{5}{9}\).
Q9. Price of 3 chairs and 2 tables is ₹4500; 5 chairs and 3 tables is ₹7000. Find price of 2 chairs and 2 tables.
Ans. Let chair = \(x\), table = \(y\). \(3x+2y=4500\); \(5x+3y=7000\). Multiply first by 3 and second by 2, subtract to get \(x=500\). Then \(3(500)+2y=4500\Rightarrow y=1500\). Required: \(2x+2y=2(500)+2(1500)=4000\).
Q10. A rectangle’s length minus breadth is 6; sum is 36. Find length and breadth.
Ans. \(x-y=6\), \(x+y=36\). Add: \(2x=42\Rightarrow x=21\). Then \(y=15\).
Q11. Find \(x,y\): \(\tfrac{x}{3}+\tfrac{y}{4}=4\), \(\tfrac{x}{2}-\tfrac{y}{4}=1\).
Ans. Multiply: \(4x+3y=48\) and \(4x-2y=8\). Subtract ⇒ \(5y=40\Rightarrow y=8\); then \(4x-16=8\Rightarrow x=6\). \((6,8)\).
Q12. Find \(x,y\): \(\tfrac{x}{3}+5y=13\), \(2x+\tfrac{y}{2}=19\).
Ans. Multiply to clear: \(x+15y=39\) and \(4x+y=38\). Eliminate: \(y=2\), then \(x=9\). \((9,2)\).
Q13. If \(x+y=5\) and \(2x+2y=10\), what relation holds between equations?
Ans. They are equivalent (same solution set); second is \(2\times\) first.
Q14. Show that \((8,6)\) is the unique common solution of \(x+y=14\) and \(x-y=2\).
Ans. Substitution/Addition gives \(x=8,y=6\). Any other ordered pair fails at least one equation.
Q15. Write any five solutions of \(x+y=7\).
Ans. \((1,6),(2,5),(3,4),(4,3),(10,-3)\) etc.
Q16. Solve \(3x-4y=7\) and \(5x+2y=3\).
Ans. Make coefficients equal: \(10x+4y=6\). Add ⇒ \(13x=13\Rightarrow x=1\), then \(y=-1\). \((1,-1)\).
Q17. Solve \(5x+7y=17\) and \(3x-2y=4\).
Ans. Multiply and add: \(10x+14y=34\) and \(21x-14y=28\) ⇒ \(31x=62\Rightarrow x=2\), \(y=1\).
Q18. Solve \(x-2y=-10\) and \(3x-5y=-12\).
Ans. Multiply first by 3: \(3x-6y=-30\). Subtract second ⇒ \(-6y+5y=-?\) gives \(y=18\); then \(x=26\). \((26,18)\).
Q19. Solve \(4x+y=34\) and \(x+4y=16\).
Ans. Multiply first by 4 and subtract ⇒ \(x=8\). Then \(y=2\). \((8,2)\).
Q20. In an exam of 60 Qs, +2 for correct and −1 for wrong. Yashwant attempted all and scored 90. How many wrong?
Ans. Let right \(x\), wrong \(y\). \(x+y=60\), \(2x-y=90\). Add ⇒ \(3x=150\Rightarrow x=50\). Hence \(y=10\) wrong.
All Textbook Exercises — Perfect Solutions
Below are clean, MathJax-formatted solutions for Practice Set 5.1, Practice Set 5.2, and Problem Set 5 from Chapter 5. Each question is in red and each solution in green so you can paste directly into your site.
Practice Set 5.1
Q1. By using variables \(x\) and \(y\) form any five linear equations in two variables.
Ans. Examples (general form \(ax+by+c=0\)):
\(3x+4y-12=0,\; 3x-4y+12=0,\; 5x+5y-6=0,\; 7x+12y-11=0,\; x-y+5=0\).
Q2. Write five solutions of \(x+y=7\).
Ans. \((1,6),(2,5),(3,4),(4,3),(0,7)\) (many more possible).
Q3(i). Solve: \(x+y=4\); \(2x-5y=1\).
Ans. \(x=3,\; y=1\).
Q3(ii). Solve: \(2x+y=5\); \(3x-y=5\).
Ans. \(x=2,\; y=1\).
Q3(iii). Solve: \(3x-5y=16\); \(x-3y=8\).
Ans. \(x=2,\; y=-2\).
Q3(iv). Solve: \(2y-x=0\); \(10x+15y=105\).
Ans. \(x=6,\; y=3\).
Q3(v). Solve: \(2x+3y+4=0\); \(x-5y=11\).
Ans. \(x=1,\; y=-2\).
Q3(vi). Solve: \(2x-7y=7\); \(3x+y=22\).
Ans. \(x=7,\; y=1\).
Let’s Recall (from page 80) — click to view
Solve: \(m+3=5\)
Ans. \(m=2\).
Solve: \(3y+8=22\)
Ans. \(3y=14\Rightarrow y=\tfrac{14}{3}\).
Solve: \(\tfrac{x}{3}=2\)
Ans. \(x=6\).
Solve: \(2p=p+\tfrac{4}{9}\)
Ans. \(p=\tfrac{4}{9}\).
Which number should be added to 5 to obtain 14?
Ans. \(9\).
Which number should be subtracted from 8 to obtain 2?
Ans. \(6\).
Practice Set 5.2
Q1. In an envelope there are some ₹5 notes and some ₹10 notes. Total amount is ₹350. Number of ₹5 notes are 10 less than twice the number of ₹10 notes. Find both counts.
Ans. Let ₹5 notes \(x\), ₹10 notes \(y\). \(5x+10y=350\Rightarrow x+2y=70\); and \(x=2y-10\). Hence \(y=20, x=30\).
Q2. Denominator is 1 less than twice the numerator. If 1 is added to numerator and denominator respectively, the ratio becomes \(3:5\). Find the fraction.
Ans. \(y=2x-1\); and \(\tfrac{x+1}{y+1}=\tfrac{3}{5}\Rightarrow 5x+5=3y+3\). So \(x=5, y=9\). Fraction \(=\tfrac{5}{9}\).
Q3. The sum of ages of Priyanka and Deepika is 34 years. Priyanka is 6 years elder. Find their ages.
Ans. \(x+y=34\), \(x-y=6\Rightarrow x=20, y=14\).
Q4. In a zoo, total lions and peacocks = 50; total legs = 140. Find numbers.
Ans. \(x+y=50\), \(4x+2y=140\Rightarrow 2x+y=70\). Subtract ⇒ \(x=20, y=30\).
Q5. Sanjay’s monthly salary: after 4 years = ₹4500, after 10 years = ₹5400. Assuming a fixed yearly increment, find original salary and yearly increment.
Ans. Let initial salary \(x\), increment \(y\). Then \(x+4y=4500\), \(x+10y=5400\). Subtract ⇒ \(6y=900\Rightarrow y=150\), and \(x=4500-4\cdot150=3900\).
Q6. Price of 3 chairs and 2 tables is ₹4500; price of 5 chairs and 3 tables is ₹7000. Find price of 2 chairs and 2 tables.
Ans. \(3x+2y=4500\), \(5x+3y=7000\Rightarrow x=500, y=1500\). Hence \(2x+2y=4000\).
Q7. Sum of digits of a two-digit number is 9. Interchanging digits gives a number 27 greater than original. Find the number.
Ans. Let tens \(y\), units \(x\). \(x+y=9\). Interchanged number \(10x+y=10y+x+27\Rightarrow 9x-9y=27\Rightarrow x-y=3\). Solve ⇒ \(x=6,y=3\). Number \(=36\).
Q8*. In \(\triangle ABC\), \(\angle A=\angle B+\angle C\) and \(\angle B:\angle C=4:5\). Find all angles.
Ans. Let \(\angle B=4k,\; \angle C=5k\Rightarrow \angle A=9k\). Sum \(=180\Rightarrow 18k=180\Rightarrow k=10\). Hence \(\angle A=90^\circ,\; \angle B=40^\circ,\; \angle C=50^\circ\).
Q9*. Divide \(560\,\text{cm}\) rope into two parts such that twice the smaller equals \(\tfrac{1}{3}\) of the larger. Find the larger.
Ans. Let smaller \(x\), larger \(560-x\). Given \(2x=\tfrac{1}{3}(560-x)\Rightarrow 6x=560-x\Rightarrow 7x=560\Rightarrow x=80\). Larger \(=480\,\text{cm}\).
Q10. Exam of 60 questions: +2 for correct, −1 for wrong. Yashwant attempted all and scored 90. How many wrong?
Ans. \(x+y=60\), \(2x-y=90\Rightarrow x=50, y=10\). 10 wrong.
Problem Set 5
Q1. Choose the correct alternatives.
(i) If \(3x+5y=9\) and \(5x+3y=7\), then \(x+y=\color{#0a8a2a}{2}\).
(ii) When 5 is subtracted from length and breadth, perimeter becomes 26 \(\Rightarrow x+y=\color{#0a8a2a}{23}\).
(iii) Ajay younger than Vijay by 5 and sum is 25: Ajay’s age = \(\color{#0a8a2a}{10}\) years.
(ii) When 5 is subtracted from length and breadth, perimeter becomes 26 \(\Rightarrow x+y=\color{#0a8a2a}{23}\).
(iii) Ajay younger than Vijay by 5 and sum is 25: Ajay’s age = \(\color{#0a8a2a}{10}\) years.
Q2(i). Solve: \(2x+y=5\); \(3x-y=5\).
Ans. \(x=2, y=1\).
Q2(ii). Solve: \(x-2y=-1\); \(2x-y=7\).
Ans. \(x=5, y=3\).
Q2(iii). Solve: \(x+y=11\); \(2x-3y=7\).
Ans. \(x=8, y=3\).
Q2(iv). Solve: \(2x+y=-2\); \(3x-y=7\).
Ans. \(x=1, y=-4\).
Q2(v). Solve: \(2x-y=5\); \(3x+2y=11\).
Ans. \(x=3, y=1\).
Q2(vi). Solve: \(x-2y=-2\); \(x+2y=10\).
Ans. \(x=4, y=3\).
Q3(i). By equating coefficients, solve: \(3x-4y=7\); \(5x+2y=3\).
Ans. \(x=1, y=-1\).
Q3(ii). Solve: \(5x+7y=17\); \(3x-2y=4\).
Ans. \(x=2, y=1\).
Q3(iii). Solve: \(x-2y=-10\); \(3x-5y=-12\).
Ans. \(x=26, y=18\).
Q3(iv). Solve: \(4x+y=34\); \(x+4y=16\).
Ans. \(x=8, y=2\).
Q4(i). Solve: \(\tfrac{x}{3}+\tfrac{y}{4}=4\); \(\tfrac{x}{2}-\tfrac{y}{4}=1\).
Ans. \(x=6, y=8\).
Q4(ii). Solve: \(\tfrac{x}{3}+5y=13\); \(2x+\tfrac{y}{2}=19\).
Ans. \(x=9, y=2\).
Q4(iii). Solve: \(\tfrac{2}{x}+\tfrac{3}{y}=13\); \(\tfrac{5}{x}+\tfrac{4}{y}=32\).
Ans. Let \(u=\tfrac{1}{x}, v=\tfrac{1}{y}\). Then \(2u+3v=13\), \(5u+4v=32\). Solve: multiply first by 4 and second by 3: \(8u+12v=52\), \(15u+12v=96\). Subtract ⇒ \(7u=44\Rightarrow u=\tfrac{44}{7}\Rightarrow x=\tfrac{7}{44}\). Substitute: \(2\tfrac{44}{7}+3v=13\Rightarrow \tfrac{88}{7}+3v=13\Rightarrow 3v=\tfrac{91-88}{7}=\tfrac{3}{7}\Rightarrow v=\tfrac{1}{7}\Rightarrow y=7\).
Q5. A two digit number is 3 more than 4 times the sum of its digits. Adding 18 gives the number with digits interchanged. Find the number.
Ans. \(x-2y=-1\) and \(x-y=2\Rightarrow x=5,y=3\). Number = 35.
Q6. The total cost of 8 books and 5 pens is ₹420 and of 5 books and 8 pens is ₹321. Find cost of 1 book and 2 pens.
Ans. \(8x+5y=420\), \(5x+8y=321\Rightarrow y=12, x=45\). Hence \(x+2y=45+24=\) ₹69.
Q7*. Ratio of incomes of two persons is 9:7; ratio of their expenses is 4:3. Each saves ₹200. Find their incomes.
Ans. Let incomes \(9k,7k\), expenses \(4m,3m\). Savings: \(9k-4m=200\) and \(7k-3m=200\). Eliminate \(m\): multiply second by 4 and first by 3 ⇒ \(28k-12m=800\) and \(27k-12m=600\). Subtract ⇒ \(k=200\). Thus incomes are ₹\(1800\) and ₹\(1400\).
Q8*. Rectangle: if length reduced by 5 and breadth increased by 3, area decreases by 8; if length reduced by 3 and breadth increased by 2, area increases by 67. Find length/breadth.
Ans. Let length \(x\), breadth \(y\). Then \((x-5)(y+3)=xy-8\Rightarrow 3x-5y= -23\). Also \((x-3)(y+2)=xy+67\Rightarrow 2x-3y=73\). Solve: multiply first by 3 and second by 5 ⇒ \(9x-15y=-69\) and \(10x-15y=365\). Subtract ⇒ \(x=434\). Then \(2(434)-3y=73\Rightarrow 868-3y=73\Rightarrow y=265\). (Large integer dimensions satisfy both equations.)
Q9*. Two places \(A,B\) are 70 km apart. Two cars: (i) same direction ⇒ meet after 7 h, (ii) opposite directions ⇒ meet after 1 h. Find their speeds.
Ans. Let speeds \(x,y\). Same direction: \(x-y=10\). Opposite: \(x+y=70\). Solve ⇒ \(x=40, y=30\) km/h.
Q10*. The sum of a two-digit number and the number obtained by interchanging its digits is 99. Find all such numbers.
Ans. Let tens \(y\), units \(x\). \(10y+x+10x+y=99\Rightarrow 11(x+y)=99\Rightarrow x+y=9\). Pairs \((x,y)=(1,8),(2,7),(3,6),\dots\). Numbers: 18, 27, 36, 45, 54, 63, 72, 81.
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