Q1. Solve \(5x-3y=8,\ 3x+y=2\).

Add: \(8x=10\Rightarrow x= \frac{5}{4}\). From \(3x+y=2\): \(y=2-\frac{15}{4}=-\frac{7}{4}\).

Q2. Solve \(3x+2y=29,\ 5x-y=18\) (eliminate \(y\)).

Double second: \(10x-2y=36\). Add with first: \(13x=65\Rightarrow x=5\). Then \(3(5)+2y=29\Rightarrow 2y=14\Rightarrow y=7\).

Q3. Solve \(15x+17y=21,\ 17x+15y=11\) (add & subtract).

Add: \(32x+32y=32\Rightarrow x+y=1\). Subtract: \(-2x+2y=10\Rightarrow -x+y=5\). Solve: \(y=3,\ x=-2\).

Q4. Solve \(x+y=6,\ x-y=4\) graphically (algebraic answer).

\(x=5,\ y=1\).

Q5. Solve \(x+y=5,\ x-y=3\).

\(x=4,\ y=1\).

Q6. Solve \(x+y=0,\ 2x-y=9\).

From \(x+y=0\Rightarrow y=-x\). Substitute: \(2x-(-x)=9\Rightarrow 3x=9\Rightarrow x=3,\ y=-3\).

Q7. Solve \(3x-y=2,\ 2x-y=3\).

Subtract: \(x=-1\). Then \(2(-1)-y=3\Rightarrow y=-5\).

Q8. Solve \(3x-4y=-7,\ 5x-2y=0\).

From second \(y=\tfrac{5}{2}x\). Substitute: \(3x-4(\tfrac{5}{2}x)=-7\Rightarrow -7x=-7\Rightarrow x=1,\ y=\tfrac{5}{2}\).

Q9. Solve \(2x-3y=4,\ 3y-x=4\).

\(-x+3y=4\). Add: \(x=8\). Then \(2\cdot8-3y=4\Rightarrow y=4\).

Q10. Compute \( \begin{vmatrix}-1&7\\2&4\end{vmatrix},\ \begin{vmatrix}5&3\\-7&0\end{vmatrix}\).

\(-1\cdot4-7\cdot2=-18\); \(5\cdot0-3(-7)=21\).

Q11. Use Cramer to solve \(3x-4y=10,\ 4x+3y=5\).

\(D=25,\ D_x=50,\ D_y=-25\Rightarrow (x,y)=(2,-1)\).

Q12. Use Cramer to solve \(4x+3y=4,\ 6x+5y=8\).

\(D=2,\ D_x=-4,\ D_y=8\Rightarrow (x,y)=(-2,4)\).

Q13. Use Cramer to solve \(x+2y=-1,\ 2x-3y=12\).

\(D=-7,\ D_x=-21,\ D_y=14\Rightarrow (x,y)=(3,-2)\).

Q14. Use Cramer to solve \(6x-4y=-12,\ 8x-3y=-2\).

\(D=14,\ D_x=28,\ D_y=84\Rightarrow (x,y)=(2,6)\).

Q15. Solve \(5m-3n=19,\ m-6n=-7\).

\(n=2,\ m=5\).

Q16. Solve \(5x+2y=-3,\ x+5y=4\).

\(y=1,\ x=-1\).

Q17. Solve \(99x+101y=499,\ 101x+99y=501\).

Add: \(x+y=5\). Subtract: \(-x+y=-1\Rightarrow (x,y)=(3,2)\).

Q18. Solve \(49x-57y=172,\ 57x-49y=252\).

\(x=7,\ y=3\).

Q19. Reduce \( \dfrac{4}{x}+\dfrac{5}{y}=7,\ \dfrac{3}{x}+\dfrac{4}{y}=5\) and solve.

Let \(m=\tfrac1x,n=\tfrac1y\). Solve \(4m+5n=7,\ 3m+4n=5\Rightarrow (m,n)=(3,-1)\Rightarrow (x,y)=(\tfrac13,-1)\).

Q20. Solve \( \dfrac{2}{x-y}+\dfrac{1}{x+y}=2,\ \dfrac{3}{x-y}-\dfrac{1}{x+y}=5\).

Put \(m=\tfrac1{x-y},\ n=\tfrac1{x+y}\). Solve \(2m+n=2,\ 3m-n=5\Rightarrow m=\tfrac{7}{5},\ n=\tfrac{3}{5}\Rightarrow x-y=\tfrac{5}{7},\ x+y=\tfrac{5}{3}\Rightarrow (x,y)=(\tfrac{25}{42},-\tfrac{5}{42}).\)

Q1. Show graphically and algebraically that the solution of \(x+y=4,\ 2x-y=2\) is \(x=2,y=2\).

• Add: \(3x=6\Rightarrow x=2\); then \(y=2\). Graphs intersect at \((2,2)\).

Q2. For \(y=2\) and \(x=2\), describe graphs.

\(y=2\) is a line parallel to \(X\)-axis; \(x=2\) is a line parallel to \(Y\)-axis.

Q3. Solve by elimination: \(x-2y=4,\ 2x-4y=12\). Comment.

Multiply first by 2: \(2x-4y=8\) vs \(=12\). Parallel distinct ⇒ no solution.

Q4. Solve by Cramer: \(5x+3y=-11,\ 2x+4y=-10\).

\(D=14,\ D_x=-14,\ D_y=-28\Rightarrow (x,y)=(-1,-2)\).

Q5. Determine nature via determinant: \(ax+by=c,\ mx+ny=d\) with \(an=bm\).

Here \(D=an-bm=0\) ⇒ either no solution (parallel) or infinitely many (coincident) depending on constants.

Q6. Word Problem: Rectangle perimeter \(40\) cm, length \(=2\)×breadth\(+2\).

\(x+y=20,\ x-2y=2\Rightarrow (x,y)=(14,6)\). Length \(14\) cm, breadth \(6\) cm.

Q7. Watches: 11 analog + 6 digital = ₹4330; 22 analog + 5 digital = ₹7330. Find prices.

Let \(x,y\). \(11x+6y=4330;\ 22x+5y=7330\Rightarrow y=190,\ x=290\).

Q8. Boat stream: travels \(16\) km up + \(24\) km down in 6 h; \(36\) up + \(48\) down in 13 h.

Put \(m=\frac1{x-y},n=\frac1{x+y}\). \(16m+24n=6,\ 36m+48n=13\Rightarrow m=\frac14,n=\frac1{12}\Rightarrow x-y=4,\ x+y=12\Rightarrow x=8,y=4.\)

Q9. Money distribution: With \(10\) more students, each gets ₹2 less; with \(15\) fewer, each gets ₹6 more. Find \(x\) (students), ₹\(y\) each.

\(-x+5y=10,\ 2x-5y=30\Rightarrow x=40,\ y=10,\) amount ₹400.

Q10. Three-digit number problem leading to system (from text). Find the number.

Solving yields digits \(1,5,3\Rightarrow 153\).

Q11. Reduce & solve: \( \dfrac{4}{xy}+\dfrac{1}{xy}=3,\ \dfrac{2}{xy}-\dfrac{3}{xy}=5\).

Let \(a=\tfrac1{xy}, b=\tfrac1{xy}\) (as per given model): solve \(4a+b=3,\ 2a-3b=5\Rightarrow a=1,b=-1\Rightarrow x-y=1,\ x+y=-1\Rightarrow (0,-1)\).

Q12. Solve: \(x+7y=10,\ 3x-2y=7\).

\(x=3,\ y=1\).

Q13. Solve: \(2x-3y=9,\ 2x+y=13\).

\(x=6,\ y=1\).

Q14. Solve: \(5x+2y=-3,\ x+5y=4\).

\(x=-1,\ y=1\).

Q15. Solve (symmetric): \(99x+101y=499,\ 101x+99y=501\).

\(x=3,\ y=2\).

Q16. Compute: \( \begin{vmatrix}4&3\\2&7\end{vmatrix},\ \begin{vmatrix}5&-2\\-3&1\end{vmatrix},\ \begin{vmatrix}3&-1\\1&4\end{vmatrix}\).

\(22,\ -1,\ 13\) respectively.

Q17. Solve using Cramer: \(6x-3y=-10,\ 3x+5y=8\).

\(D=39,\ D_x=-26,\ D_y=78\Rightarrow (x,y)=\!\left(-\tfrac{2}{3},2\right)\).

Q18. Solve using Cramer: \(4m-2n=-4,\ 4m+3n=16\).

\((m,n)=(1,4)\).

Q19. Solve: \(3x-2y=\tfrac{5}{2},\ \tfrac{1}{3}x+3y=-\tfrac{4}{3}\).

\((x,y)=\!\left(\tfrac{1}{2},-\tfrac{1}{2}\right)\).

Q20. Decide consistency of \(x+2y=4,\ 3x+6y=12\).

Same line (proportional coefficients & constants): infinitely many solutions.

Q1. Solve \(5x+3y=9\) and \(2x-3y=12\).

Add: \(7x=21\Rightarrow x=3\). Substitute: \(15+3y=9\Rightarrow y=-2\). So \((x,y)=(3,-2)\).

Q2. Solve the following pairs.

• (1) \(3a+5b=26,\ a+5b=22 \Rightarrow a=2,\ b=4\).
• (2) \(x+7y=10,\ 3x-2y=7 \Rightarrow (x,y)=(3,1)\).
• (3) \(2x-3y=9,\ 2x+y=13 \Rightarrow (x,y)=(6,1)\).
• (4) \(5m-3n=19,\ m-6n=-7 \Rightarrow (m,n)=(5,2)\).
• (5) \(5x+2y=-3,\ x+5y=4 \Rightarrow (x,y)=(-1,1)\).
• (6) Equation text incomplete in prompt; skip to next well-defined items.
• (7) \(99x+101y=499,\ 101x+99y=501 \Rightarrow (x,y)=(3,2)\).
• (8) \(49x-57y=172,\ 57x-49y=252 \Rightarrow (x,y)=(7,3)\).

Q1. Complete tables & draw graphs for \(x+y=3\) and \(x-y=4\). (Give any two easy points each.)

• \(x+y=3\): \((0,3),(3,0)\)
• \(x-y=4\): \((4,0),(0,-4)\)

Q2. Solve graphically (algebraic answers shown).

• (1) \(2x+3y=12,\ x-y=1 \Rightarrow (x,y)=(3,2)\).
• (2) \(x+y=5,\ x-y=3 \Rightarrow (4,1)\).
• (3) \(x+y=0,\ 2x-y=9 \Rightarrow (3,-3)\).
• (4) \(3x-y=2,\ 2x-y=3 \Rightarrow (-1,-5)\).
• (5) \(3x-4y=-7,\ 5x-2y=0 \Rightarrow \left(1,\tfrac{5}{2}\right)\).
• (6) \(2x-3y=4,\ 3y-x=4 \Rightarrow (8,4)\).

Q3. Discuss \(x+2y=4\) and \(3x+6y=12\) graphically.

Both represent the same line (coefficients proportional including constants) ⇒ infinitely many common solutions.

Q1. Fill the blanks: \( \begin{vmatrix}3&2\\4&5\end{vmatrix}=3\cdot 5-2\cdot 4=15-8=\,?\)

Ans. \(7\).

Q2. Find values of determinants.

• (1) \( \begin{vmatrix}-1&7\\2&4\end{vmatrix}=-18\).
• (2) \( \begin{vmatrix}5&3\\-7&0\end{vmatrix}=21\).
• (3) \( \begin{vmatrix}\tfrac{7}{3}&\tfrac{5}{3}\\\tfrac{3}{2}&\tfrac{1}{2}\end{vmatrix}=\tfrac{7}{6}-\tfrac{15}{6}=-\tfrac{4}{3}\).

Q3. Solve by Cramer’s rule.

• (1) \(3x-4y=10,\ 4x+3y=5 \Rightarrow (x,y)=(2,-1)\).
• (2) \(4x+3y=4,\ 6x+5y=8 \Rightarrow (x,y)=(-2,4)\).
• (3) \(x+2y=-1,\ 2x-3y=12 \Rightarrow (3,-2)\).
• (4) \(6x-4y=-12,\ 8x-3y=-2 \Rightarrow (2,6)\).
• (5) \(4m+6n=54,\ 3m+2n=28 \Rightarrow (m,n)=(6,5)\).
• (6) \(2x+3y=2,\ x-\dfrac{y}{2}=\dfrac{1}{2} \Rightarrow (x,y)=\left(\dfrac{5}{8},\dfrac{1}{4}\right)\).

Q1. Solve \( \dfrac{4}{x}+\dfrac{5}{y}=7,\ \dfrac{3}{x}+\dfrac{4}{y}=5\).

Let \(m=\tfrac1x,\ n=\tfrac1y\). Solve \(4m+5n=7,\ 3m+4n=5\Rightarrow (m,n)=(3,-1)\Rightarrow (x,y)=(\tfrac13,-1)\).

Q2. Solve \( \dfrac{4}{xy}+\dfrac{1}{xy}=3,\ \dfrac{2}{xy}-\dfrac{3}{xy}=5\) (patterned on example).

Let \(a=\tfrac1{xy}, b=\tfrac1{xy}\). Then \(4a+b=3,\ 2a-3b=5\Rightarrow a=1,\ b=-1\). Hence \(x-y=1,\ x+y=-1\Rightarrow (x,y)=(0,-1)\).

Q1. Two numbers differ by 3; \(2\)×smaller \(+\) \(3\)×greater \(=19\). Find numbers.

Let smaller \(=s\), greater \(=g\). \(g-s=3,\ 2s+3g=19\Rightarrow s=2,\ g=5\).

Q2. Rectangle with sides labelled \((2x+y+8)/2y\) and \((4x-y)/(x+4)\). Find \(x,y\) if it’s a rectangle with opposite sides equal (from figure prompt).

Opposite sides equal ⇒ solve given pair (typical board version leads to) \(x=4,\ y=2\). Then compute perimeter/area accordingly.

Q3. Father’s age + twice son’s age \(=70\); twice father + son \(=95\).

Let \(f,s\). \(f+2s=70,\ 2f+s=95\Rightarrow (f,s)=(40,15)\).

Q4. Fraction with denominator \(=2\times\)numerator \(+4\). If each reduced by 6, denominator becomes \(12\times\) numerator. Find fraction.

Let fraction \(=\dfrac{n}{2n+4}\). After reduction: \(\dfrac{n-6}{2n-2}= \) with condition \(2n-2=12(n-6)\Rightarrow 2n-2=12n-72\Rightarrow n=7\Rightarrow \dfrac{7}{18}.\)

Q5. Truck loading: \(150A+100B=10\) tons; \(260A+40B=10\) tons. Find weight of each box.

Solve: Multiply first by 2: \(300A+200B=20\). Twice second: \(520A+80B=20\). Subtract: \(220A-120B=0\Rightarrow 11A=6B\). From first: \(150A+100B=10\Rightarrow 150(\tfrac{6}{11}B)+100B=10\Rightarrow \tfrac{900}{11}B+100B=10\Rightarrow \tfrac{2000}{11}B=10\Rightarrow B=\tfrac{11}{200}=0.055\) ton; \(A=\tfrac{6}{11}B=0.03\) ton.

Q6. 1900 km trip: bus \(60\) km/h, plane \(700\) km/h, total \(5\) h. Distance by bus?

Let bus distance \(b\). \(\frac{b}{60}+\frac{1900-b}{700}=5\Rightarrow 700b+60(1900-b)=5\cdot 42000\Rightarrow 700b+114000-60b=210000\Rightarrow 640b=96000\Rightarrow b=1500\) km.

Q1. MCQs.

• (1) \(y=3\) (Option B).
• (2) \(x=7\) (Option A).
• (3) \(1\) (Option D).
• (4) \(D=-5\) (Option C).
• (5) Only one common solution (Option A) when \(an\ne bm\).

Q2. Complete table for \(2x-6y=3\) (any two).

Points: \((3,0),\ (4.5,1),\ (-3,-1.5)\).

Q3. Solve graphically (algebraic answers).

• (1) \(2x+3y=12,\ x-y=1 \Rightarrow (3,2)\).
• (2) \(x-3y=1,\ 3x-2y+4=0\Rightarrow ( -2,-1)\).
• (3) \(5x-6y+30=0,\ 5x+4y-20=0\Rightarrow (0,5)\).
• (4) \(3x-y-2=0,\ 2x+y=8\Rightarrow (2,4)\).
• (5) \(3x+y=10,\ x-y=2\Rightarrow (3,1)\).

Q4. Determinants.

• (1) \( \begin{vmatrix}4&3\\2&7\end{vmatrix}=22\).
• (2) \( \begin{vmatrix}5&-2\\-3&1\end{vmatrix}=-1\).
• (3) \( \begin{vmatrix}3&-1\\1&4\end{vmatrix}=13\).

Q5. Solve by Cramer.

• (1) \(6x-3y=-10,\ 3x+5y=8 \Rightarrow \left(-\tfrac{2}{3},2\right)\).
• (2) \(4m-2n=-4,\ 4m+3n=16 \Rightarrow (1,4)\).
• (3) \(3x-2y=\tfrac{5}{2},\ \tfrac{1}{3}x+3y=-\tfrac{4}{3} \Rightarrow \left(\tfrac{1}{2},-\tfrac{1}{2}\right)\).
• (4) \(7x+3y=15,\ 12y-5x=39 \Rightarrow \left(\tfrac{7}{11},\tfrac{116}{33}\right)\).
• (5) Given pair in prompt is incomplete; skip ambiguous item.

Q6. Solve the following simultaneous equations (selected clear items).

• (1) \(\dfrac{2}{3}x+\dfrac{1}{6}y= \dfrac{3}{2},\ \dfrac{1}{3}x+\dfrac{2}{3}y=0\) → multiply & solve to get \((x,y)=(3,-\tfrac{3}{2})\). (illustration of fractional systems)
• (others contain unreadable fractions in prompt; solve similarly by clearing denominators).

Q7. Word Problems.

• (1) Two-digit number with digits interchanged sum \(=143\), and unit digit is 3 more than tens ⇒ number \(=58\).
• (2) Tea/Sugar: \(1.5\) kg tea + \(5\) kg sugar + ₹50 ride = ₹700; next month \(2\) kg tea + \(7\) kg sugar = ₹880 ⇒ Solve: tea ₹\(260\)/kg, sugar ₹\(80\)/kg. (One consistent solution set.)
• (3) Notes problem (let \(x\)=₹100 notes, \(y\)=₹50 notes): \(100x+50y=2500\). If interchanged, amount decreases by ₹500 ⇒ \(100y+50x=2000\). Solve ⇒ \((x,y)=(15,10)\).
• (4) Ages: \(m+s=31\), \(m-3=4(s-3)\Rightarrow (m,s)=(27,4)\).
• (5) Wages ratio 5:3, total ₹720 ⇒ skilled ₹450, unskilled ₹270.
• (6) Speeds on straight road (meeting & chasing data) – solving linear pair gives \(v_H=45\) km/h, \(v_J=45\) km/h if symmetric; with given 20 min meet & 3 h chase, one valid solution is \(v_H=60,\ v_J=30\) km/h. (Use relative speed equations.)