Q1. Solve \(x+4=9\).

Ans: \(x=9-4=5.\)

Q2. Solve \(x-2=7\).

Ans: \(x=7+2=9.\)

Q3. Solve \(\dfrac{x}{3}=4\).

Ans: \(x=4\times3=12.\)

Q4. Solve \(4x=24\).

Ans: \(x=24/4=6.\)

Q5. Is \(x=7\) a solution of \(2x-1=13\)?

Ans: LHS \(=2\times7-1=13=\) RHS → Yes.

Q6. Solve \(9x-4=6x+29\).

Ans: \(9x-6x=29+4\Rightarrow3x=33\Rightarrow x=11.\)

Q7. Solve \(5(x-3)=3(x+2)\).

Ans: \(5x-15=3x+6\Rightarrow2x=21\Rightarrow x=10.5.\)

Q8. Solve \(\dfrac{2}{3}+\;5a=4\).

Ans: \(5a=4-\tfrac{2}{3}=\tfrac{10}{3}\Rightarrow a=\tfrac{2}{3}.\)

Q9. Solve \(\dfrac{x-7}{2}=\dfrac{5}{4}(?)\) — example from book: \(\dfrac{x-7}{2}=\dfrac{5}{4}\).

Ans: Multiply by 4: \(2(x-7)=5\Rightarrow2x-14=5\Rightarrow2x=19\Rightarrow x=9.5.\)

Q10. Solve \(8(2m-1)=2(2m+3)\).

Ans: \(16m-8=4m+6\Rightarrow12m=14\Rightarrow m=7/6.\)

Q11. What is the solution of \(17p-2=49\)?

Ans: \(17p=51\Rightarrow p=3.\)

Q12. Solve \(2m+7=9\).

Ans: \(2m=2\Rightarrow m=1.\)

Q13. Solve \(3x+12=2x-4\).

Ans: \(x=-16.\)

Q14. Is \(x=3\) a solution of \(9x=81\)?

Ans: \(9\times3=27\ne81\) → No. (Correct solution \(x=9\)).

Q15. Solve \(\dfrac{9}{8}x+1=10\) (as in practice).

Ans: \(\dfrac{9}{8}x=9\Rightarrow x=8.\)

Q16. Solve \(\dfrac{y}{7}\cdot\dfrac{4}{3}=2\) (i.e. \(\dfrac{4y}{21}=2\)).

Ans: \(4y=42\Rightarrow y=10.5.\)

Q17. Solve \(\dfrac{13x-5}{?}= \) — practice: \(13x-5=\dfrac{3}{2}\) (example)

Ans (if \(13x-5=\tfrac{3}{2}\)): \(13x=\tfrac{13}{2}\Rightarrow x=\tfrac{1}{2}.\)

Q18. Solve \(3(y+8)=10(y-4)+8\).

Ans: \(3y+24=10y-40+8=10y-32\Rightarrow -7y=-56\Rightarrow y=8.\)

Q19. Solve \(\dfrac{x-9}{5}=\dfrac{5}{7}\).

Ans: \(x-9=5\times\dfrac{5}{7}=\dfrac{25}{7}\Rightarrow x=9+\dfrac{25}{7}=\dfrac{88}{7}.\)

Q20. Solve \(\dfrac{y-4}{3}+3y=4\).

Ans: Multiply by 3: \(y-4+9y=12\Rightarrow10y=16\Rightarrow y=1.6.\)

Q1. Solve \(5(x-3)=3(x+4)\) (worked)

Ans: \(5x-15=3x+12\Rightarrow2x=27\Rightarrow x=\tfrac{27}{2}=13.5.\)

Q2. Solve \(9x-4=6x+29\) (worked).

Ans: \(3x=33\Rightarrow x=11.\)

Q3. Solve \( \dfrac{2}{3} +5a = 4\) by two methods (brief).

Ans: Subtract \(2/3\): \(5a=4-\tfrac{2}{3}=\tfrac{10}{3}\Rightarrow a=\tfrac{2}{3}\). Or multiply by 3: \(2+15a=12\Rightarrow a=\tfrac{2}{3}.\)

Q4. Solve \(\dfrac{x-7}{2}=\dfrac{5}{4}\) (book example solved).

Ans: Multiply by 4: \(2(x-7)=5\Rightarrow2x=19\Rightarrow x=\tfrac{19}{2}=9.5.\)

Q5. Solve \(8(2m-1)=2(2m+3)\) (worked).

Ans: \(16m-8=4m+6\Rightarrow12m=14\Rightarrow m=\tfrac{7}{6}.\)

Q6. Solve \(2(x+5)=15\).

Ans: \(x+5=7.5\Rightarrow x=2.5.\)

Q7. Solve \(4(x-1)+3=2x+7\).

Ans: \(4x-4+3=2x+7\Rightarrow2x=8\Rightarrow x=4.\)

Q8. Solve \( \dfrac{3x}{4}-2=1\).

Ans: \(\dfrac{3x}{4}=3\Rightarrow x=4.\)

Q9. Solve \(7x+5=3x+21\).

Ans: \(4x=16\Rightarrow x=4.\)

Q10. Solve \(5x-3(x+2)=7\).

Ans: \(5x-3x-6=7\Rightarrow2x=13\Rightarrow x=6.5.\)

Q11. Solve \( \dfrac{2x+1}{3}=5\).

Ans: \(2x+1=15\Rightarrow2x=14\Rightarrow x=7.\)

Q12. Solve \(3(2y-1)=4(y+5)\).

Ans: \(6y-3=4y+20\Rightarrow2y=23\Rightarrow y=11.5.\)

Q13. Solve \( \dfrac{4}{5}x - \dfrac{1}{2} = 3\).

Ans: \(\dfrac{4}{5}x = 3.5 = \dfrac{7}{2}\Rightarrow x=\dfrac{7}{2}\cdot\dfrac{5}{4}=\dfrac{35}{8}=4.375.\)

Q14. Solve \( (x+2)(x-1)=0\) (factorization → solutions).

Ans: \(x=-2\) or \(x=1.\)

Q15. If \(2x+3=11\), find \(x\) and check.

Ans: \(x=4\). Check: \(2\cdot4+3=11\) OK.

Q16. Solve \( \dfrac{x+2}{5}+\dfrac{x-1}{3}=4\).

Ans: Multiply 15: \(3(x+2)+5(x-1)=60\Rightarrow3x+6+5x-5=60\Rightarrow8x+1=60\Rightarrow x=\tfrac{59}{8}=7.375.\)

Q17. Solve \(6-2x=10\).

Ans: \(-2x=4\Rightarrow x=-2.\)

Q18. Solve \( \dfrac{5x-3}{2}=7\).

Ans: \(5x-3=14\Rightarrow5x=17\Rightarrow x=\tfrac{17}{5}=3.4.\)

Q19. Solve \(x/4 + x/2 = 9\).

Ans: Multiply 4: \(x+2x=36\Rightarrow3x=36\Rightarrow x=12.\)

Q20. Solve \(2(3x-4)=18\).

Ans: \(6x-8=18\Rightarrow6x=26\Rightarrow x=\dfrac{13}{3}=4.\overline{3}.\)

Q1. Solve and check: \(5(x-3)=3(x+4)\). Show every step.

Ans: \(5x-15=3x+12\Rightarrow5x-3x=12+15\Rightarrow2x=27\Rightarrow x=\dfrac{27}{2}.\) Check: LHS \(=5(\tfrac{27}{2}-3)=5(\tfrac{21}{2})=\tfrac{105}{2}\). RHS \(=3(\tfrac{27}{2}+4)=3(\tfrac{35}{2})=\tfrac{105}{2}\). OK.

Q2. Solve \(3(y+8)=10(y-4)+8\) stepwise.

Ans: \(3y+24=10y-40+8=10y-32\Rightarrow24+32=10y-3y\Rightarrow56=7y\Rightarrow y=8.\)

Q3. Solve \(\dfrac{x-9}{5}=\dfrac{5}{7}\) and express as mixed or fraction.

Ans: \(x-9=\dfrac{25}{7}\Rightarrow x=\dfrac{25}{7}+9=\dfrac{25+63}{7}=\dfrac{88}{7}=12\dfrac{4}{7}.\)

Q4. Solve \( \dfrac{2x+1}{3}+ \dfrac{x-2}{4}=5\) (clear denominators).

Ans: Multiply 12: \(4(2x+1)+3(x-2)=60\Rightarrow8x+4+3x-6=60\Rightarrow11x-2=60\Rightarrow11x=62\Rightarrow x=\tfrac{62}{11}.\)

Q5. Word problem: "My friend's age is \( \tfrac{x}{2}+5\). If friend is 12, find my age \(x\)." (worked)

Ans: \(\tfrac{x}{2}+5=12\Rightarrow\tfrac{x}{2}=7\Rightarrow x=14.\)

Q6. Solve \( \dfrac{4}{5}x - \dfrac{2}{3} = 1\). (clear fractions)

Ans: Multiply by 15: \(12x-10=15\Rightarrow12x=25\Rightarrow x=\tfrac{25}{12}.\)

Q7. Solve and interpret: \( \dfrac{y-4}{3}+3y=4\).

Ans: Multiply 3: \(y-4+9y=12\Rightarrow10y=16\Rightarrow y=1.6.\)

Q8. Solve \( (x-1)(x+2)=x^2+x-2\); reduce to linear if possible.

Ans: Expand LHS \(=x^2+x-2\) → same as RHS → identity: every real \(x\) satisfies. So infinite solutions: all real numbers.

Q9. Solve: \( \dfrac{x+2}{5}+\dfrac{x-1}{3}=4\) (worked earlier).

Ans: Multiply 15: \(3(x+2)+5(x-1)=60\Rightarrow8x+1=60\Rightarrow x=\tfrac{59}{8}.\)

Q10. Solve the equation from textbook: \(\dfrac{8m-1}{1} = 2(2m+3)\) (verify given example).

Ans: \(8m-1=4m+6\Rightarrow4m=7\Rightarrow m=\tfrac{7}{4}.\)

Q11. Translate and solve: "My grandmother's age = \(4x+10\). My friend's age is \(x/2+5\). If friend = 12, find grandmother's age."

Ans: From earlier \(x=14\). Grandmother \(=4x+10=4\cdot14+10=66\) years.

Q12. Solve: \( \dfrac{x}{4} + \dfrac{x}{2} + \dfrac{x}{8}=7\).

Ans: LCM 8: \(2x+4x+x=56\Rightarrow7x=56\Rightarrow x=8.\)

Q13. If \(3x+2=5x-10\), find \(x\) and check.

Ans: \(-2x=-12\Rightarrow x=6\). Check: \(3\cdot6+2=20, 5\cdot6-10=20\) OK.

Q14. Solve: \( \dfrac{3x+2}{4} - \dfrac{x-1}{2} = 3\).

Ans: Multiply 4: \(3x+2-2(x-1)=12\Rightarrow3x+2-2x+2=12\Rightarrow x+4=12\Rightarrow x=8.\)

Q15. Find \(x\) if \(2(3x-4)=3(x+2)+x\).

Ans: \(6x-8=3x+6+x\Rightarrow6x-8=4x+6\Rightarrow2x=14\Rightarrow x=7.\)

Q16. Word problem: Joseph & brother weights sum 63, Joseph is twice brother → find weights.

Ans: If brother \(=x\), Joseph \(=2x\). \(x+2x=63\Rightarrow3x=63\Rightarrow x=21\). Joseph = 42 kg, brother = 21 kg.

Q17. Word problem: numerator = denominator +5; after adding 4 to both, fraction = \(\tfrac{6}{5}\). Find fraction.

Ans: Denominator \(=x\), numerator \(=x+5\). \(\dfrac{x+5+4}{x+4}=\dfrac{6}{5}\Rightarrow\dfrac{x+9}{x+4}=\dfrac{6}{5}\Rightarrow5(x+9)=6(x+4)\Rightarrow5x+45=6x+24\Rightarrow x=21.\) Fraction \(=\dfrac{26}{21}.\)

Q18. Solve: Ratna–Rafik money problem (worked; textbook sketch).

Ans (short): Let Rafik \(=x\). Ratna = \(3x+200\). After transfer 300: Ratna \(=3x+200-300=3x-100\), Rafik \(=x+300\). Given \( \dfrac{3x-100}{x+300}=\dfrac{7}{4}\). Solve: \(4(3x-100)=7(x+300)\Rightarrow12x-400=7x+2100\Rightarrow5x=2500\Rightarrow x=500.\) Rafik had Rs. 500 initially.

Q19. If three consecutive numbers sum to between 45 and 54, find them (example request).

Ans: Let numbers \(n,n+1,n+2\). Sum \(=3n+3\). Solve \(45<3n+3<54\Rightarrow14< n <17\). Integer \(n=15,16\). For \(n=15\) sum=48 (works), numbers 15,16,17. For \(n=16\) sum=51 (works), numbers 16,17,18.

Q20. Average problem: player scores 180,257, needs score \(x\) so avg =230. Find \(x\).

Ans: \(\dfrac{180+257+x}{3}=230\Rightarrow437+x=690\Rightarrow x=253.\)

1.(1) For equation \(x-4=3\), check \(x=-1,7,-7\).

Ans: LHS with \(x=-1\): \(-1-4=-5\ne3\). \(x=7\): \(7-4=3\) → Yes. \(x=-7\): \(-7-4=-11\ne3\). So only \(x=7\).

1.(2) For \(9m=81\), check \(m=3,9,-3\).

Ans: \(m=3\): \(9\cdot3=27\ne81\). \(m=9\): \(9\cdot9=81\) → Yes. \(m=-3\): \(9(-3)=-27\ne81\). So only \(m=9\).

1.(3) For \(2a+4=0\), check \(a=2,-2,1\).

Ans: \(a=2\): \(4+4=8\ne0\). \(a=-2\): \(-4+4=0\) → Yes. \(a=1\): \(2+4=6\ne0\).

1.(4) For \(3-y=4\), check \(y=-1,1,2\).

Ans: \(y=-1\): \(3-(-1)=4\) → Yes. \(y=1\): \(3-1=2\ne4\). \(y=2\): \(3-2=1\ne4\).

2.(1) Solve \(17p-2=49\).

Ans: \(17p=51\Rightarrow p=3.\)

2.(2) Solve \(2m+7=9\).

Ans: \(2m=2\Rightarrow m=1.\)

2.(3) Solve \(3x+12=2x-4\).

Ans: \(3x-2x=-4-12\Rightarrow x=-16.\)

2.(4) Solve \(5(x-3)=3(x+2)\).

Ans: \(5x-15=3x+6\Rightarrow2x=21\Rightarrow x=\tfrac{21}{2}=10.5.\)

2.(5) Solve \(\dfrac{9}{8}x+1=10\) (as in practice set text: "9/8 x +1 =10").

Ans: \(\dfrac{9}{8}x=9\Rightarrow x=9\times\dfrac{8}{9}=8.\)

2.(6) Solve \( \dfrac{y}{7}\cdot\dfrac{4}{3}=2\) (interpreted as \(\dfrac{4y}{21}=2\)).

Ans: \(4y=42\Rightarrow y=10.5.\)

2.(7) Solve \(13x-5=\dfrac{3}{2}\) (or if intended \(13x-5=3/2\)).

Ans: \(13x=\tfrac{3}{2}+5=\tfrac{13}{2}\Rightarrow x=\tfrac{1}{2}.\)

2.(8) Solve \(3(y+8)=10(y-4)+8\).

Ans: \(3y+24=10y-40+8=10y-32\Rightarrow -7y=-56\Rightarrow y=8.\)

2.(9) Solve \(\dfrac{x-9}{5}=\dfrac{5}{7}\) (repeat).

Ans: \(x=\dfrac{88}{7}.\)

2.(10) Solve \(\dfrac{y-4}{3}+3y=4\) (repeat).

Ans: \(y=1.6.\)

2.(11) Solve (complex fraction from prompt) — interpret: \( \dfrac{b+1}{4} + \dfrac{1}{2} + (b) ??? =21\).

Ans: The prompt has unclear formatting. If equation is \( \dfrac{b+1}{4} + \dfrac{b}{2} =21\), multiply 4: \(b+1+2b=84\Rightarrow3b=83\Rightarrow b=\tfrac{83}{3}\). If user meant different, paste exact equation and I'll solve precisely.

1. Mother is 25 years older than son. After 8 years ratio son:mother \(=4:9\). Find son's age now.

Ans: Let son \(=x\). Mother \(=x+25\). After 8 years: \(\dfrac{x+8}{x+25+8}=\dfrac{4}{9}\). \(\dfrac{x+8}{x+33}=\dfrac{4}{9}\Rightarrow9(x+8)=4(x+33)\Rightarrow9x+72=4x+132\Rightarrow5x=60\Rightarrow x=12.\)

2. Denominator greater than numerator by 12. If numerator decreased by 2 and denominator increased by 7, new fraction = \(1/2\). Find original fraction.

Ans: Let numerator \(=n\), denom \(=n+12\). \(\dfrac{n-2}{n+12+7}=\dfrac{1}{2}\Rightarrow\dfrac{n-2}{n+19}=\dfrac{1}{2}\Rightarrow2n-4=n+19\Rightarrow n=23.\) Original fraction \(=\dfrac{23}{35}.\)

(alternate) If adding 4 to both gives fraction \(6/5\) (second part of problem), solve:

If \(\dfrac{n+4}{n+16}=\dfrac{6}{5}\Rightarrow5(n+4)=6(n+16)\Rightarrow5n+20=6n+96\Rightarrow n=-76\) (not a positive denom) — so probably separate parts. Follow original text carefully.

3. Ratio copper:zinc = 13:7. In 700 g brass, weight of zinc = ?

Ans: Total parts = 13+7=20. Zinc = \(7/20\times700=7\times35=245\) g.

4. Find three consecutive whole numbers whose sum is >45 and <54.

Ans: Let numbers \(n,n+1,n+2\). Sum \(=3n+3\). Solve \(45<3n+3<54\Rightarrow14

5. Two-digit number: tens digit twice units digit. Sum of number and its digit-swapped form =66. Find number.

Ans: Let units = \(u\). Tens = \(2u\). Number \(=10(2u)+u=21u\). Swapped = \(10u+2u=12u\). Sum \(21u+12u=33u=66\Rightarrow u=2.\) Number \(=21\cdot2=42.\)

6. Tickets of Rs.200 and Rs.100: number of 200 tickets is 20 more than 100 tickets. Total amount Rs.37000. Find number of 100 tickets.

Ans: Let number of 100-tickets = \(x\). 200-tickets = \(x+20\). Revenue \(=100x+200(x+20)=100x+200x+4000=300x+4000=37000\Rightarrow300x=33000\Rightarrow x=110.\)

7. Of three consecutive natural numbers, five times smallest = 9 more than four times greatest. Find numbers.

Ans: Let numbers \(n,n+1,n+2\). Equation: \(5n=4(n+2)+9\Rightarrow5n=4n+8+9=4n+17\Rightarrow n=17.\) Numbers: 17,18,19.

8. Bicycle problem: Raju sold to Amit at 8% profit. Amit repaired Rs.54 and sold to Nikhil for Rs.1134 with no profit/loss. Find Raju's cost price.

Ans: Let Raju's CP = \(C\). He sold to Amit at 8% profit → Amit paid \(1.08C\). Amit spent 54 on repair, so his total cost = \(1.08C+54\). He sold to Nikhil at no profit → selling price = Amit's total cost = \(1134\). So \(1.08C+54=1134\Rightarrow1.08C=1080\Rightarrow C=1080/1.08=1000.\) Raju's CP = Rs.1000.

9. Cricket average problem (repeat): need 253 in third match.

Ans: \(x=253\).

10. Age-relationship (Practice 12.2 Q10): Sudhir present age = 5 + 3V (V=Viru). Anil = half of Sudhir. If (Sudhir+Viru)/(3*Anil) = 5/6, find Viru's age.

Ans: Let Viru \(=v\). Sudhir \(=3v+5\). Anil \(=(3v+5)/2\). Given \(\dfrac{(3v+5)+v}{3\cdot\frac{3v+5}{2}}=\dfrac{5}{6}.\) Simplify: \(\dfrac{4v+5}{\tfrac{9v+15}{2}}=\dfrac{5}{6}\Rightarrow \dfrac{2(4v+5)}{9v+15}=\dfrac{5}{6}.\) Cross-multiply: \(12(4v+5)=5(9v+15)\Rightarrow48v+60=45v+75\Rightarrow3v=15\Rightarrow v=5.\)