Chapter 3 usually deals with Linear Equations in Two Variables, where we learn how to solve systems of linear equations by different methods like Substitution Method, Elimination Method, and Graphical Method.
Exercise 3.1 – Linear Equations in Two Variables
Q1. Solve the following system of equations:
x+y=10x + y = 10x+y=10x−y=4x – y = 4x−y=4
Solution:
Using Elimination Method, add both equations:(x+y)+(x−y)=10+4(x + y) + (x – y) = 10 + 4(x+y)+(x−y)=10+42x=142x = 142x=14x=142=7x = \frac{14}{2} = 7x=214=7
Substitute x=7x = 7x=7 into the first equation:7+y=107 + y = 107+y=10y=10−7=3y = 10 – 7 = 3y=10−7=3
Answer: x=7x = 7x=7, y=3y = 3y=3
Q2. Solve the system of equations:
3x+2y=163x + 2y = 163x+2y=162x−y=42x – y = 42x−y=4
Solution:
Using Substitution Method, solve for yyy in the second equation:2x−y=42x – y = 42x−y=4y=2x−4y = 2x – 4y=2x−4
Substitute y=2x−4y = 2x – 4y=2x−4 into the first equation:3x+2(2x−4)=163x + 2(2x – 4) = 163x+2(2x−4)=163x+4x−8=163x + 4x – 8 = 163x+4x−8=167x=247x = 247x=24x=247x = \frac{24}{7}x=724
Now, substitute x=247x = \frac{24}{7}x=724 into y=2x−4y = 2x – 4y=2x−4:y=2(247)−4=487−287=207y = 2\left(\frac{24}{7}\right) – 4 = \frac{48}{7} – \frac{28}{7} = \frac{20}{7}y=2(724)−4=748−728=720
Answer: x=247x = \frac{24}{7}x=724, y=207y = \frac{20}{7}y=720
Q3. Solve the system of equations:
x−2y=3x – 2y = 3x−2y=33x+y=123x + y = 123x+y=12
Solution:
Using Elimination Method, multiply the second equation by 2:3x+y=12becomes6x+2y=243x + y = 12 \quad \text{becomes} \quad 6x + 2y = 243x+y=12becomes6x+2y=24
Now, subtract the first equation from the modified second equation:(6x+2y)−(x−2y)=24−3(6x + 2y) – (x – 2y) = 24 – 3(6x+2y)−(x−2y)=24−35x=215x = 215x=21x=215x = \frac{21}{5}x=521
Substitute x=215x = \frac{21}{5}x=521 into the first equation:215−2y=3\frac{21}{5} – 2y = 3521−2y=3−2y=3−215=155−215=−65-2y = 3 – \frac{21}{5} = \frac{15}{5} – \frac{21}{5} = \frac{-6}{5}−2y=3−521=515−521=5−6y=35y = \frac{3}{5}y=53
Answer: x=215x = \frac{21}{5}x=521, y=35y = \frac{3}{5}y=53
Q4. Solve the system of equations:
2x−y=52x – y = 52x−y=54x+y=94x + y = 94x+y=9
Solution:
Using Elimination Method, add both equations:(2x−y)+(4x+y)=5+9(2x – y) + (4x + y) = 5 + 9(2x−y)+(4x+y)=5+96x=146x = 146x=14x=146=73x = \frac{14}{6} = \frac{7}{3}x=614=37
Substitute x=73x = \frac{7}{3}x=37 into the first equation:2(73)−y=52\left(\frac{7}{3}\right) – y = 52(37)−y=5143−y=5\frac{14}{3} – y = 5314−y=5y=143−5=143−153=−13y = \frac{14}{3} – 5 = \frac{14}{3} – \frac{15}{3} = \frac{-1}{3}y=314−5=314−315=3−1
Answer: x=73x = \frac{7}{3}x=37, y=−13y = \frac{-1}{3}y=3−1
Exercise 3.2 – Graphical Method
Q5. Solve the following system of equations by graphing:
x+y=6x + y = 6x+y=6x−y=2x – y = 2x−y=2
Solution:
To graph these equations, rewrite them in slope-intercept form y=mx+cy = mx + cy=mx+c:
- x+y=6⇒y=6−xx + y = 6 \Rightarrow y = 6 – xx+y=6⇒y=6−x
- x−y=2⇒y=x−2x – y = 2 \Rightarrow y = x – 2x−y=2⇒y=x−2
Plot both lines on a graph, and the point of intersection gives the solution.
After plotting the lines, we see the lines intersect at (4,2)(4, 2)(4,2).
Answer: x=4x = 4x=4, y=2y = 2y=2
Additional 40 Questions with Solutions
Here are 40 additional questions from Linear Equations in Two Variables, and their detailed solutions:
Q6. Solve the system of equations:
x+y=5x + y = 5x+y=52x−y=42x – y = 42x−y=4
Solution:
Step 1: Add both equations to eliminate yyy:(x+y)+(2x−y)=5+4(x + y) + (2x – y) = 5 + 4(x+y)+(2x−y)=5+43x=9⇒x=93=33x = 9 \quad \Rightarrow \quad x = \frac{9}{3} = 33x=9⇒x=39=3
Step 2: Substitute x=3x = 3x=3 into the first equation:3+y=5⇒y=5−3=23 + y = 5 \quad \Rightarrow \quad y = 5 – 3 = 23+y=5⇒y=5−3=2
Answer: x=3x = 3x=3, y=2y = 2y=2
Q7. Solve the system of equations:
4x−y=94x – y = 94x−y=93x+2y=73x + 2y = 73x+2y=7
Solution:
Step 1: Solve the first equation for yyy:4x−y=9⇒y=4x−94x – y = 9 \quad \Rightarrow \quad y = 4x – 94x−y=9⇒y=4x−9
Step 2: Substitute y=4x−9y = 4x – 9y=4x−9 into the second equation:3x+2(4x−9)=73x + 2(4x – 9) = 73x+2(4x−9)=73x+8x−18=73x + 8x – 18 = 73x+8x−18=711x=25⇒x=251111x = 25 \quad \Rightarrow \quad x = \frac{25}{11}11x=25⇒x=1125
Step 3: Substitute x=2511x = \frac{25}{11}x=1125 into y=4x−9y = 4x – 9y=4x−9:y=4(2511)−9=10011−9911=111y = 4\left(\frac{25}{11}\right) – 9 = \frac{100}{11} – \frac{99}{11} = \frac{1}{11}y=4(1125)−9=11100−1199=111
Answer: x=2511x = \frac{25}{11}x=1125, y=111y = \frac{1}{11}y=111
Q8. Solve the system of equations:
x+3y=8x + 3y = 8x+3y=82x−y=42x – y = 42x−y=4
Solution:
Step 1: Solve the second equation for yyy:2x−y=4⇒y=2x−42x – y = 4 \quad \Rightarrow \quad y = 2x – 42x−y=4⇒y=2x−4
Step 2: Substitute y=2x−4y = 2x – 4y=2x−4 into the first equation:x+3(2x−4)=8x + 3(2x – 4) = 8x+3(2x−4)=8x+6x−12=8x + 6x – 12 = 8x+6x−12=87x=20⇒x=2077x = 20 \quad \Rightarrow \quad x = \frac{20}{7}7x=20⇒x=720
Step 3: Substitute x=207x = \frac{20}{7}x=720 into y=2x−4y = 2x – 4y=2x−4:y=2(207)−4=407−287=127y = 2\left(\frac{20}{7}\right) – 4 = \frac{40}{7} – \frac{28}{7} = \frac{12}{7}y=2(720)−4=740−728=712
Answer: x=207x = \frac{20}{7}x=720, y=127y = \frac{12}{7}y=712
Q9. Solve the system of equations:
x−y=5x – y = 5x−y=53x+y=113x + y = 113x+y=11
Solution:
Step 1: Add both equations to eliminate yyy:(x−y)+(3x+y)=5+11(x – y) + (3x + y) = 5 + 11(x−y)+(3x+y)=5+114x=16⇒x=164=44x = 16 \quad \Rightarrow \quad x = \frac{16}{4} = 44x=16⇒x=416=4
Step 2: Substitute x=4x = 4x=4 into the first equation:4−y=5⇒y=−14 – y = 5 \quad \Rightarrow \quad y = -14−y=5⇒y=−1
Answer: x=4x = 4x=4, y=−1y = -1y=−1
Q10. Solve the system of equations:
2x+3y=102x + 3y = 102x+3y=104x−y=94x – y = 94x−y=9
Solution:
Step 1: Solve the second equation for yyy:4x−y=9⇒y=4x−94x – y = 9 \quad \Rightarrow \quad y = 4x – 94x−y=9⇒y=4x−9
Step 2: Substitute y=4x−9y = 4x – 9y=4x−9 into the first equation:2x+3(4x−9)=102x + 3(4x – 9) = 102x+3(4x−9)=102x+12x−27=102x + 12x – 27 = 102x+12x−27=1014x=37⇒x=371414x = 37 \quad \Rightarrow \quad x = \frac{37}{14}14x=37⇒x=1437
Step 3: Substitute x=3714x = \frac{37}{14}x=1437 into y=4x−9y = 4x – 9y=4x−9:y=4(3714)−9=14814−12614=2214=117y = 4\left(\frac{37}{14}\right) – 9 = \frac{148}{14} – \frac{126}{14} = \frac{22}{14} = \frac{11}{7}y=4(1437)−9=14148−14126=1422=711
Answer: x=3714x = \frac{37}{14}x=1437, y=117y = \frac{11}{7}y=711
Q11. Solve the system of equations:
3x+2y=73x + 2y = 73x+2y=7x−2y=1x – 2y = 1x−2y=1
Solution:
Step 1: Solve the second equation for xxx:x=2y+1x = 2y + 1x=2y+1
Step 2: Substitute x=2y+1x = 2y + 1x=2y+1 into the first equation:3(2y+1)+2y=73(2y + 1) + 2y = 73(2y+1)+2y=76y+3+2y=76y + 3 + 2y = 76y+3+2y=78y=4⇒y=48=128y = 4 \quad \Rightarrow \quad y = \frac{4}{8} = \frac{1}{2}8y=4⇒y=84=21
Step 3: Substitute y=12y = \frac{1}{2}y=21 into x=2y+1x = 2y + 1x=2y+1:x=2(12)+1=1+1=2x = 2\left(\frac{1}{2}\right) + 1 = 1 + 1 = 2x=2(21)+1=1+1=2
Answer: x=2x = 2x=2, y=12y = \frac{1}{2}y=21
Q12. Solve the system of equations:
5x+2y=125x + 2y = 125x+2y=123x−4y=73x – 4y = 73x−4y=7
Solution:
Step 1: Multiply the first equation by 2 and the second by 1 to eliminate yyy:10x+4y=2410x + 4y = 2410x+4y=243x−4y=73x – 4y = 73x−4y=7
Step 2: Add both equations to eliminate yyy:(10x+4y)+(3x−4y)=24+7(10x + 4y) + (3x – 4y) = 24 + 7(10x+4y)+(3x−4y)=24+713x=31⇒x=311313x = 31 \quad \Rightarrow \quad x = \frac{31}{13}13x=31⇒x=1331
Step 3: Substitute x=3113x = \frac{31}{13}x=1331 into 5x+2y=125x + 2y = 125x+2y=12:5(3113)+2y=125\left(\frac{31}{13}\right) + 2y = 125(1331)+2y=1215513+2y=12\frac{155}{13} + 2y = 1213155+2y=122y=12−15513=15613−15513=1132y = 12 – \frac{155}{13} = \frac{156}{13} – \frac{155}{13} = \frac{1}{13}2y=12−13155=13156−13155=131y=126y = \frac{1}{26}y=261
Answer: x=3113x = \frac{31}{13}x=1331, y=126y = \frac{1}{26}y=261
Q13. Solve the system of equations:
x+4y=10x + 4y = 10x+4y=10 3x−2y=43x – 2y = 43x−2y=4
Solution:
Step 1: Solve the first equation for xxx:x=10−4yx = 10 – 4yx=10−4y
Step 2: Substitute x=10−4yx = 10 – 4yx=10−4y into the second equation:3(10−4y)−2y=43(10 – 4y) – 2y = 43(10−4y)−2y=4 30−12y−2y=430 – 12y – 2y = 430−12y−2y=4 30−14y=4⇒−14y=−26⇒y=2614=13730 – 14y = 4 \quad \Rightarrow \quad -14y = -26 \quad \Rightarrow \quad y = \frac{26}{14} = \frac{13}{7}30−14y=4⇒−14y=−26⇒y=1426=713
Step 3: Substitute y=137y = \frac{13}{7}y=713 into x=10−4yx = 10 – 4yx=10−4y:x=10−4(137)=10−527=707−527=187x = 10 – 4\left(\frac{13}{7}\right) = 10 – \frac{52}{7} = \frac{70}{7} – \frac{52}{7} = \frac{18}{7}x=10−4(713)=10−752=770−752=718
Answer: x=187x = \frac{18}{7}x=718, y=137y = \frac{13}{7}y=713
Q14. Solve the system of equations:
2x−3y=52x – 3y = 52x−3y=5 4x+y=94x + y = 94x+y=9
Solution:
Step 1: Solve the second equation for yyy:y=9−4xy = 9 – 4xy=9−4x
Step 2: Substitute y=9−4xy = 9 – 4xy=9−4x into the first equation:2x−3(9−4x)=52x – 3(9 – 4x) = 52x−3(9−4x)=5 2x−27+12x=52x – 27 + 12x = 52x−27+12x=5 14x=32⇒x=3214=16714x = 32 \quad \Rightarrow \quad x = \frac{32}{14} = \frac{16}{7}14x=32⇒x=1432=716
Step 3: Substitute x=167x = \frac{16}{7}x=716 into y=9−4xy = 9 – 4xy=9−4x:y=9−4(167)=9−647=637−647=−17y = 9 – 4\left(\frac{16}{7}\right) = 9 – \frac{64}{7} = \frac{63}{7} – \frac{64}{7} = \frac{-1}{7}y=9−4(716)=9−764=763−764=7−1
Answer: x=167x = \frac{16}{7}x=716, y=−17y = \frac{-1}{7}y=7−1
Q15. Solve the system of equations:
x+2y=7x + 2y = 7x+2y=7 3x+y=83x + y = 83x+y=8
Solution:
Step 1: Solve the second equation for yyy:y=8−3xy = 8 – 3xy=8−3x
Step 2: Substitute y=8−3xy = 8 – 3xy=8−3x into the first equation:x+2(8−3x)=7x + 2(8 – 3x) = 7x+2(8−3x)=7 x+16−6x=7x + 16 – 6x = 7x+16−6x=7 −5x=−9⇒x=95-5x = -9 \quad \Rightarrow \quad x = \frac{9}{5}−5x=−9⇒x=59
Step 3: Substitute x=95x = \frac{9}{5}x=59 into y=8−3xy = 8 – 3xy=8−3x:y=8−3(95)=8−275=405−275=135y = 8 – 3\left(\frac{9}{5}\right) = 8 – \frac{27}{5} = \frac{40}{5} – \frac{27}{5} = \frac{13}{5}y=8−3(59)=8−527=540−527=513
Answer: x=95x = \frac{9}{5}x=59, y=135y = \frac{13}{5}y=513
Q16. Solve the system of equations:
x+3y=10x + 3y = 10x+3y=10 2x−y=52x – y = 52x−y=5
Solution:
Step 1: Solve the second equation for yyy:y=2x−5y = 2x – 5y=2x−5
Step 2: Substitute y=2x−5y = 2x – 5y=2x−5 into the first equation:x+3(2x−5)=10x + 3(2x – 5) = 10x+3(2x−5)=10 x+6x−15=10x + 6x – 15 = 10x+6x−15=10 7x=25⇒x=2577x = 25 \quad \Rightarrow \quad x = \frac{25}{7}7x=25⇒x=725
Step 3: Substitute x=257x = \frac{25}{7}x=725 into y=2x−5y = 2x – 5y=2x−5:y=2(257)−5=507−357=157y = 2\left(\frac{25}{7}\right) – 5 = \frac{50}{7} – \frac{35}{7} = \frac{15}{7}y=2(725)−5=750−735=715
Answer: x=257x = \frac{25}{7}x=725, y=157y = \frac{15}{7}y=715
Q17. Solve the system of equations:
4x−y=104x – y = 104x−y=10 5x+y=145x + y = 145x+y=14
Solution:
Step 1: Add both equations to eliminate yyy:(4x−y)+(5x+y)=10+14(4x – y) + (5x + y) = 10 + 14(4x−y)+(5x+y)=10+14 9x=24⇒x=249=839x = 24 \quad \Rightarrow \quad x = \frac{24}{9} = \frac{8}{3}9x=24⇒x=924=38
Step 2: Substitute x=83x = \frac{8}{3}x=38 into the first equation:4(83)−y=104\left(\frac{8}{3}\right) – y = 104(38)−y=10 323−y=10\frac{32}{3} – y = 10332−y=10 −y=10−323=303−323=−23-y = 10 – \frac{32}{3} = \frac{30}{3} – \frac{32}{3} = \frac{-2}{3}−y=10−332=330−332=3−2 y=23y = \frac{2}{3}y=32
Answer: x=83x = \frac{8}{3}x=38, y=23y = \frac{2}{3}y=32
Q18. Solve the system of equations:
x+y=6x + y = 6x+y=6 x−2y=4x – 2y = 4x−2y=4
Solution:
Step 1: Add both equations to eliminate yyy:(x+y)+(x−2y)=6+4(x + y) + (x – 2y) = 6 + 4(x+y)+(x−2y)=6+4 2x−y=102x – y = 102x−y=10
Step 2: Solve for yyy in terms of xxx from the modified second equation:y=2x−10y = 2x – 10y=2x−10
Step 3: Substitute y=2x−10y = 2x – 10y=2x−10 into the first equation:x+(2x−10)=6x + (2x – 10) = 6x+(2x−10)=6 3x−10=63x – 10 = 63x−10=6 3x=16⇒x=1633x = 16 \quad \Rightarrow \quad x = \frac{16}{3}3x=16⇒x=316
Step 4: Substitute x=163x = \frac{16}{3}x=316 into y=2x−10y = 2x – 10y=2x−10:y=2(163)−10=323−303=23y = 2\left(\frac{16}{3}\right) – 10 = \frac{32}{3} – \frac{30}{3} = \frac{2}{3}y=2(316)−10=332−330=32
Answer: x=163x = \frac{16}{3}x=316, y=23y = \frac{2}{3}y=32
Q19. Solve the system of equations:
3x+4y=113x + 4y = 113x+4y=11 2x−y=52x – y = 52x−y=5
Solution:
Step 1: Solve the second equation for yyy:y=2x−5y = 2x – 5y=2x−5
Step 2: Substitute y=2x−5y = 2x – 5y=2x−5 into the first equation:3x+4(2x−5)=113x + 4(2x – 5) = 113x+4(2x−5)=11 3x+8x−20=113x + 8x – 20 = 113x+8x−20=11 11x=31⇒x=311111x = 31 \quad \Rightarrow \quad x = \frac{31}{11}11x=31⇒x=1131
Step 3: Substitute x=3111x = \frac{31}{11}x=1131 into y=2x−5y = 2x – 5y=2x−5:y=2(3111)−5=6211−5511=711y = 2\left(\frac{31}{11}\right) – 5 = \frac{62}{11} – \frac{55}{11} = \frac{7}{11}y=2(1131)−5=1162−1155=117
Answer: x=3111x = \frac{31}{11}x=1131, y=711y = \frac{7}{11}y=117
Q20. Solve the system of equations:
x+y=9x + y = 9x+y=9 3x−y=53x – y = 53x−y=5
Solution:
Step 1: Add both equations to eliminate yyy:(x+y)+(3x−y)=9+5(x + y) + (3x – y) = 9 + 5(x+y)+(3x−y)=9+5 4x=14⇒x=144=724x = 14 \quad \Rightarrow \quad x = \frac{14}{4} = \frac{7}{2}4x=14⇒x=414=27
Step 2: Substitute x=72x = \frac{7}{2}x=27 into the first equation:72+y=9⇒y=9−72=182−72=112\frac{7}{2} + y = 9 \quad \Rightarrow \quad y = 9 – \frac{7}{2} = \frac{18}{2} – \frac{7}{2} = \frac{11}{2}27+y=9⇒y=9−27=218−27=211
Answer: x=72x = \frac{7}{2}x=27, y=112y = \frac{11}{2}y=211
Q21. Solve the system of equations:
x+y=12x + y = 12x+y=12 x−y=4x – y = 4x−y=4
Solution:
Step 1: Add both equations to eliminate yyy:(x+y)+(x−y)=12+4(x + y) + (x – y) = 12 + 4(x+y)+(x−y)=12+4 2x=16⇒x=162=82x = 16 \quad \Rightarrow \quad x = \frac{16}{2} = 82x=16⇒x=216=8
Step 2: Substitute x=8x = 8x=8 into the first equation:8+y=12⇒y=12−8=48 + y = 12 \quad \Rightarrow \quad y = 12 – 8 = 48+y=12⇒y=12−8=4
Answer: x=8x = 8x=8, y=4y = 4y=4
Q22. Solve the system of equations:
2x+3y=132x + 3y = 132x+3y=13 4x−y=34x – y = 34x−y=3
Solution:
Step 1: Solve the second equation for yyy:4x−y=3⇒y=4x−34x – y = 3 \quad \Rightarrow \quad y = 4x – 34x−y=3⇒y=4x−3
Step 2: Substitute y=4x−3y = 4x – 3y=4x−3 into the first equation:2x+3(4x−3)=132x + 3(4x – 3) = 132x+3(4x−3)=13 2x+12x−9=132x + 12x – 9 = 132x+12x−9=13 14x=22⇒x=2214=11714x = 22 \quad \Rightarrow \quad x = \frac{22}{14} = \frac{11}{7}14x=22⇒x=1422=711
Step 3: Substitute x=117x = \frac{11}{7}x=711 into y=4x−3y = 4x – 3y=4x−3:y=4(117)−3=447−217=237y = 4\left(\frac{11}{7}\right) – 3 = \frac{44}{7} – \frac{21}{7} = \frac{23}{7}y=4(711)−3=744−721=723
Answer: x=117x = \frac{11}{7}x=711, y=237y = \frac{23}{7}y=723
Q23. Solve the system of equations:
x+y=7x + y = 7x+y=7 2x−y=82x – y = 82x−y=8
Solution:
Step 1: Add both equations to eliminate yyy:(x+y)+(2x−y)=7+8(x + y) + (2x – y) = 7 + 8(x+y)+(2x−y)=7+8 3x=15⇒x=153=53x = 15 \quad \Rightarrow \quad x = \frac{15}{3} = 53x=15⇒x=315=5
Step 2: Substitute x=5x = 5x=5 into the first equation:5+y=7⇒y=7−5=25 + y = 7 \quad \Rightarrow \quad y = 7 – 5 = 25+y=7⇒y=7−5=2
Answer: x=5x = 5x=5, y=2y = 2y=2
Q24. Solve the system of equations:
x−y=1x – y = 1x−y=1 2x+y=92x + y = 92x+y=9
Solution:
Step 1: Add both equations to eliminate yyy:(x−y)+(2x+y)=1+9(x – y) + (2x + y) = 1 + 9(x−y)+(2x+y)=1+9 3x=10⇒x=1033x = 10 \quad \Rightarrow \quad x = \frac{10}{3}3x=10⇒x=310
Step 2: Substitute x=103x = \frac{10}{3}x=310 into the first equation:103−y=1⇒y=103−1=103−33=73\frac{10}{3} – y = 1 \quad \Rightarrow \quad y = \frac{10}{3} – 1 = \frac{10}{3} – \frac{3}{3} = \frac{7}{3}310−y=1⇒y=310−1=310−33=37
Answer: x=103x = \frac{10}{3}x=310, y=73y = \frac{7}{3}y=37
Q25. Solve the system of equations:
3x+y=103x + y = 103x+y=10 5x−2y=45x – 2y = 45x−2y=4
Solution:
Step 1: Solve the first equation for yyy:y=10−3xy = 10 – 3xy=10−3x
Step 2: Substitute y=10−3xy = 10 – 3xy=10−3x into the second equation:5x−2(10−3x)=45x – 2(10 – 3x) = 45x−2(10−3x)=4 5x−20+6x=45x – 20 + 6x = 45x−20+6x=4 11x=24⇒x=241111x = 24 \quad \Rightarrow \quad x = \frac{24}{11}11x=24⇒x=1124
Step 3: Substitute x=2411x = \frac{24}{11}x=1124 into y=10−3xy = 10 – 3xy=10−3x:y=10−3(2411)=10−7211=11011−7211=3811y = 10 – 3\left(\frac{24}{11}\right) = 10 – \frac{72}{11} = \frac{110}{11} – \frac{72}{11} = \frac{38}{11}y=10−3(1124)=10−1172=11110−1172=1138
Answer: x=2411x = \frac{24}{11}x=1124, y=3811y = \frac{38}{11}y=1138
Q26. Solve the system of equations:
2x−y=12x – y = 12x−y=1 3x+y=113x + y = 113x+y=11
Solution:
Step 1: Add both equations to eliminate yyy:(2x−y)+(3x+y)=1+11(2x – y) + (3x + y) = 1 + 11(2x−y)+(3x+y)=1+11 5x=12⇒x=1255x = 12 \quad \Rightarrow \quad x = \frac{12}{5}5x=12⇒x=512
Step 2: Substitute x=125x = \frac{12}{5}x=512 into the first equation:2(125)−y=1⇒245−y=12\left(\frac{12}{5}\right) – y = 1 \quad \Rightarrow \quad \frac{24}{5} – y = 12(512)−y=1⇒524−y=1 −y=1−245=55−245=−195-y = 1 – \frac{24}{5} = \frac{5}{5} – \frac{24}{5} = \frac{-19}{5}−y=1−524=55−524=5−19 y=195y = \frac{19}{5}y=519
Answer: x=125x = \frac{12}{5}x=512, y=195y = \frac{19}{5}y=519
Q27. Solve the system of equations:
x−2y=4x – 2y = 4x−2y=4 3x+y=93x + y = 93x+y=9
Solution:
Step 1: Solve the first equation for xxx:x=2y+4x = 2y + 4x=2y+4
Step 2: Substitute x=2y+4x = 2y + 4x=2y+4 into the second equation:3(2y+4)+y=93(2y + 4) + y = 93(2y+4)+y=9 6y+12+y=96y + 12 + y = 96y+12+y=9 7y=−3⇒y=−377y = -3 \quad \Rightarrow \quad y = \frac{-3}{7}7y=−3⇒y=7−3
Step 3: Substitute y=−37y = \frac{-3}{7}y=7−3 into x=2y+4x = 2y + 4x=2y+4:x=2(−37)+4=−67+4=−67+287=227x = 2\left(\frac{-3}{7}\right) + 4 = \frac{-6}{7} + 4 = \frac{-6}{7} + \frac{28}{7} = \frac{22}{7}x=2(7−3)+4=7−6+4=7−6+728=722
Answer: x=227x = \frac{22}{7}x=722, y=−37y = \frac{-3}{7}y=7−3
Q28. Solve the system of equations:
5x+3y=155x + 3y = 155x+3y=15 2x−y=32x – y = 32x−y=3
Solution:
Step 1: Solve the second equation for yyy:y=2x−3y = 2x – 3y=2x−3
Step 2: Substitute y=2x−3y = 2x – 3y=2x−3 into the first equation:5x+3(2x−3)=155x + 3(2x – 3) = 155x+3(2x−3)=15 5x+6x−9=155x + 6x – 9 = 155x+6x−9=15 11x=24⇒x=241111x = 24 \quad \Rightarrow \quad x = \frac{24}{11}11x=24⇒x=1124
Step 3: Substitute x=2411x = \frac{24}{11}x=1124 into y=2x−3y = 2x – 3y=2x−3:y=2(2411)−3=4811−3311=1511y = 2\left(\frac{24}{11}\right) – 3 = \frac{48}{11} – \frac{33}{11} = \frac{15}{11}y=2(1124)−3=1148−1133=1115
Answer: x=2411x = \frac{24}{11}x=1124, y=1511y = \frac{15}{11}y=1115
Q29. Solve the system of equations:
x+y=8x + y = 8x+y=8 2x−y=52x – y = 52x−y=5
Solution:
Step 1: Add both equations to eliminate yyy:(x+y)+(2x−y)=8+5(x + y) + (2x – y) = 8 + 5(x+y)+(2x−y)=8+5 3x=13⇒x=1333x = 13 \quad \Rightarrow \quad x = \frac{13}{3}3x=13⇒x=313
Step 2: Substitute x=133x = \frac{13}{3}x=313 into the first equation:133+y=8⇒y=8−133=243−133=113\frac{13}{3} + y = 8 \quad \Rightarrow \quad y = 8 – \frac{13}{3} = \frac{24}{3} – \frac{13}{3} = \frac{11}{3}313+y=8⇒y=8−313=324−313=311
Answer: x=133x = \frac{13}{3}x=313, y=113y = \frac{11}{3}y=311
Q30. Solve the system of equations:
3x+y=73x + y = 73x+y=7 4x−2y=64x – 2y = 64x−2y=6
Solution:
Step 1: Solve the first equation for yyy:y=7−3xy = 7 – 3xy=7−3x
Step 2: Substitute y=7−3xy = 7 – 3xy=7−3x into the second equation:4x−2(7−3x)=64x – 2(7 – 3x) = 64x−2(7−3x)=6 4x−14+6x=64x – 14 + 6x = 64x−14+6x=6 10x=20⇒x=2010=210x = 20 \quad \Rightarrow \quad x = \frac{20}{10} = 210x=20⇒x=1020=2
Step 3: Substitute x=2x = 2x=2 into y=7−3xy = 7 – 3xy=7−3x:y=7−3(2)=7−6=1y = 7 – 3(2) = 7 – 6 = 1y=7−3(2)=7−6=1
Answer: x=2x = 2x=2, y=1y = 1y=1
Q31. Solve the system of equations:
4x−3y=74x – 3y = 74x−3y=7 2x+y=52x + y = 52x+y=5
Solution:
Step 1: Solve the second equation for yyy:y=5−2xy = 5 – 2xy=5−2x
Step 2: Substitute y=5−2xy = 5 – 2xy=5−2x into the first equation:4x−3(5−2x)=74x – 3(5 – 2x) = 74x−3(5−2x)=7 4x−15+6x=74x – 15 + 6x = 74x−15+6x=7 10x=22⇒x=2210=11510x = 22 \quad \Rightarrow \quad x = \frac{22}{10} = \frac{11}{5}10x=22⇒x=1022=511
Step 3: Substitute x=115x = \frac{11}{5}x=511 into y=5−2xy = 5 – 2xy=5−2x:y=5−2(115)=5−225=255−225=35y = 5 – 2\left(\frac{11}{5}\right) = 5 – \frac{22}{5} = \frac{25}{5} – \frac{22}{5} = \frac{3}{5}y=5−2(511)=5−522=525−522=53
Answer: x=115x = \frac{11}{5}x=511, y=35y = \frac{3}{5}y=53
Q32. Solve the system of equations:
3x+2y=83x + 2y = 83x+2y=8 5x−y=75x – y = 75x−y=7
Solution:
Step 1: Solve the second equation for yyy:y=5x−7y = 5x – 7y=5x−7
Step 2: Substitute y=5x−7y = 5x – 7y=5x−7 into the first equation:3x+2(5x−7)=83x + 2(5x – 7) = 83x+2(5x−7)=8 3x+10x−14=83x + 10x – 14 = 83x+10x−14=8 13x=22⇒x=221313x = 22 \quad \Rightarrow \quad x = \frac{22}{13}13x=22⇒x=1322
Step 3: Substitute x=2213x = \frac{22}{13}x=1322 into y=5x−7y = 5x – 7y=5x−7:y=5(2213)−7=11013−9113=1913y = 5\left(\frac{22}{13}\right) – 7 = \frac{110}{13} – \frac{91}{13} = \frac{19}{13}y=5(1322)−7=13110−1391=1319
Answer: x=2213x = \frac{22}{13}x=1322, y=1913y = \frac{19}{13}y=1319
Q33. Solve the system of equations:
x−2y=3x – 2y = 3x−2y=3 3x+4y=123x + 4y = 123x+4y=12
Solution:
Step 1: Solve the first equation for xxx:x=2y+3x = 2y + 3x=2y+3
Step 2: Substitute x=2y+3x = 2y + 3x=2y+3 into the second equation:3(2y+3)+4y=123(2y + 3) + 4y = 123(2y+3)+4y=12 6y+9+4y=126y + 9 + 4y = 126y+9+4y=12 10y=3⇒y=31010y = 3 \quad \Rightarrow \quad y = \frac{3}{10}10y=3⇒y=103
Step 3: Substitute y=310y = \frac{3}{10}y=103 into x=2y+3x = 2y + 3x=2y+3:x=2(310)+3=610+3=610+3010=3610=185x = 2\left(\frac{3}{10}\right) + 3 = \frac{6}{10} + 3 = \frac{6}{10} + \frac{30}{10} = \frac{36}{10} = \frac{18}{5}x=2(103)+3=106+3=106+1030=1036=518
Answer: x=185x = \frac{18}{5}x=518, y=310y = \frac{3}{10}y=103
Q34. Solve the system of equations:
2x−y=42x – y = 42x−y=4 x+2y=10x + 2y = 10x+2y=10
Solution:
Step 1: Solve the second equation for xxx:x=10−2yx = 10 – 2yx=10−2y
Step 2: Substitute x=10−2yx = 10 – 2yx=10−2y into the first equation:2(10−2y)−y=42(10 – 2y) – y = 42(10−2y)−y=4 20−4y−y=420 – 4y – y = 420−4y−y=4 20−5y=4⇒−5y=−16⇒y=16520 – 5y = 4 \quad \Rightarrow \quad -5y = -16 \quad \Rightarrow \quad y = \frac{16}{5}20−5y=4⇒−5y=−16⇒y=516
Step 3: Substitute y=165y = \frac{16}{5}y=516 into x=10−2yx = 10 – 2yx=10−2y:x=10−2(165)=10−325=505−325=185x = 10 – 2\left(\frac{16}{5}\right) = 10 – \frac{32}{5} = \frac{50}{5} – \frac{32}{5} = \frac{18}{5}x=10−2(516)=10−532=550−532=518
Answer: x=185x = \frac{18}{5}x=518, y=165y = \frac{16}{5}y=516
Q35. Solve the system of equations:
x+3y=15x + 3y = 15x+3y=15 4x−2y=84x – 2y = 84x−2y=8
Solution:
Step 1: Solve the first equation for xxx:x=15−3yx = 15 – 3yx=15−3y
Step 2: Substitute x=15−3yx = 15 – 3yx=15−3y into the second equation:4(15−3y)−2y=84(15 – 3y) – 2y = 84(15−3y)−2y=8 60−12y−2y=860 – 12y – 2y = 860−12y−2y=8 60−14y=8⇒−14y=−52⇒y=5214=26760 – 14y = 8 \quad \Rightarrow \quad -14y = -52 \quad \Rightarrow \quad y = \frac{52}{14} = \frac{26}{7}60−14y=8⇒−14y=−52⇒y=1452=726
Step 3: Substitute y=267y = \frac{26}{7}y=726 into x=15−3yx = 15 – 3yx=15−3y:x=15−3(267)=15−787=1057−787=277x = 15 – 3\left(\frac{26}{7}\right) = 15 – \frac{78}{7} = \frac{105}{7} – \frac{78}{7} = \frac{27}{7}x=15−3(726)=15−778=7105−778=727
Answer: x=277x = \frac{27}{7}x=727, y=267y = \frac{26}{7}y=726
Q36. Solve the system of equations:
x+2y=5x + 2y = 5x+2y=5 3x−y=43x – y = 43x−y=4
Solution:
Step 1: Solve the first equation for xxx:x=5−2yx = 5 – 2yx=5−2y
Step 2: Substitute x=5−2yx = 5 – 2yx=5−2y into the second equation:3(5−2y)−y=43(5 – 2y) – y = 43(5−2y)−y=4 15−6y−y=415 – 6y – y = 415−6y−y=4 15−7y=4⇒−7y=−11⇒y=11715 – 7y = 4 \quad \Rightarrow \quad -7y = -11 \quad \Rightarrow \quad y = \frac{11}{7}15−7y=4⇒−7y=−11⇒y=711
Step 3: Substitute y=117y = \frac{11}{7}y=711 into x=5−2yx = 5 – 2yx=5−2y:x=5−2(117)=5−227=357−227=137x = 5 – 2\left(\frac{11}{7}\right) = 5 – \frac{22}{7} = \frac{35}{7} – \frac{22}{7} = \frac{13}{7}x=5−2(711)=5−722=735−722=713
Answer: x=137x = \frac{13}{7}x=713, y=117y = \frac{11}{7}y=711
Q37. Solve the system of equations:
x−y=4x – y = 4x−y=4 x+y=10x + y = 10x+y=10
Solution:
Step 1: Add both equations to eliminate yyy:(x−y)+(x+y)=4+10(x – y) + (x + y) = 4 + 10(x−y)+(x+y)=4+10 2x=14⇒x=142=72x = 14 \quad \Rightarrow \quad x = \frac{14}{2} = 72x=14⇒x=214=7
Step 2: Substitute x=7x = 7x=7 into the first equation:7−y=4⇒y=7−4=37 – y = 4 \quad \Rightarrow \quad y = 7 – 4 = 37−y=4⇒y=7−4=3
Answer: x=7x = 7x=7, y=3y = 3y=3
Q38. Solve the system of equations:
x+4y=12x + 4y = 12x+4y=12 2x−y=32x – y = 32x−y=3
Solution:
Step 1: Solve the second equation for yyy:y=2x−3y = 2x – 3y=2x−3
Step 2: Substitute y=2x−3y = 2x – 3y=2x−3 into the first equation:x+4(2x−3)=12x + 4(2x – 3) = 12x+4(2x−3)=12 x+8x−12=12x + 8x – 12 = 12x+8x−12=12 9x=24⇒x=249=839x = 24 \quad \Rightarrow \quad x = \frac{24}{9} = \frac{8}{3}9x=24⇒x=924=38
Step 3: Substitute x=83x = \frac{8}{3}x=38 into y=2x−3y = 2x – 3y=2x−3:y=2(83)−3=163−93=73y = 2\left(\frac{8}{3}\right) – 3 = \frac{16}{3} – \frac{9}{3} = \frac{7}{3}y=2(38)−3=316−39=37
Answer: x=83x = \frac{8}{3}x=38, y=73y = \frac{7}{3}y=37
Q39. Solve the system of equations:
3x+2y=103x + 2y = 103x+2y=10 x−y=1x – y = 1x−y=1
Solution:
Step 1: Solve the second equation for xxx:x=y+1x = y + 1x=y+1
Step 2: Substitute x=y+1x = y + 1x=y+1 into the first equation:3(y+1)+2y=103(y + 1) + 2y = 103(y+1)+2y=10 3y+3+2y=103y + 3 + 2y = 103y+3+2y=10 5y+3=10⇒5y=7⇒y=755y + 3 = 10 \quad \Rightarrow \quad 5y = 7 \quad \Rightarrow \quad y = \frac{7}{5}5y+3=10⇒5y=7⇒y=57
Step 3: Substitute y=75y = \frac{7}{5}y=57 into x=y+1x = y + 1x=y+1:x=75+1=75+55=125x = \frac{7}{5} + 1 = \frac{7}{5} + \frac{5}{5} = \frac{12}{5}x=57+1=57+55=512
Answer: x=125x = \frac{12}{5}x=512, y=75y = \frac{7}{5}y=57
Q40. Solve the system of equations:
4x+y=154x + y = 154x+y=15 2x−3y=42x – 3y = 42x−3y=4
Solution:
Step 1: Solve the first equation for yyy:y=15−4xy = 15 – 4xy=15−4x
Step 2: Substitute y=15−4xy = 15 – 4xy=15−4x into the second equation:2x−3(15−4x)=42x – 3(15 – 4x) = 42x−3(15−4x)=4 2x−45+12x=42x – 45 + 12x = 42x−45+12x=4 14x−45=4⇒14x=49⇒x=4914=7214x – 45 = 4 \quad \Rightarrow \quad 14x = 49 \quad \Rightarrow \quad x = \frac{49}{14} = \frac{7}{2}14x−45=4⇒14x=49⇒x=1449=27
Step 3: Substitute x=72x = \frac{7}{2}x=27 into y=15−4xy = 15 – 4xy=15−4x:y=15−4(72)=15−14=1y = 15 – 4\left(\frac{7}{2}\right) = 15 – 14 = 1y=15−4(27)=15−14=1
Answer: x=72x = \frac{7}{2}x=27, y=1y = 1y=1
Summary of Answers:
- x=8,y=4x = 8, y = 4x=8,y=4
- x=117,y=237x = \frac{11}{7}, y = \frac{23}{7}x=711,y=723
- x=5,y=2x = 5, y = 2x=5,y=2
- x=103,y=73x = \frac{10}{3}, y = \frac{7}{3}x=310,y=37
- x=2411,y=3811x = \frac{24}{11}, y = \frac{38}{11}x=1124,y=1138
- x=125,y=195x = \frac{12}{5}, y = \frac{19}{5}x=512,y=519
- x=227,y=−37x = \frac{22}{7}, y = \frac{-3}{7}x=722,y=7−3
- x=2411,y=1511x = \frac{24}{11}, y = \frac{15}{11}x=1124,y=1115
- x=133,y=113x = \frac{13}{3}, y = \frac{11}{3}x=313,y=311
- x=2,y=1x = 2, y = 1x=2,y=1
- x=115,y=35x = \frac{11}{5}, y = \frac{3}{5}x=511,y=53
- x=2213,y=1913x = \frac{22}{13}, y = \frac{19}{13}x=1322,y=1319
- x=185,y=310x = \frac{18}{5}, y = \frac{3}{10}x=518,y=103
- x=185,y=165x = \frac{18}{5}, y = \frac{16}{5}x=518,y=516
- x=277,y=267x = \frac{27}{7}, y = \frac{26}{7}x=727,y=726
- x=137,y=117x = \frac{13}{7}, y = \frac{11}{7}x=713,y=711
- x=135,y=85x = \frac{13}{5}, y = \frac{8}{5}x=513,y=58
- x=2,y=1x = 2, y = 1x=2,y=1
- x=7,y=3x = 7, y = 3x=7,y=3
- x=277,y=267x = \frac{27}{7}, y = \frac{26}{7}x=727,y=726
- x=75,y=3x = \frac{7}{5}, y = 3x=57,y=3
- x=75,y=55x = \frac{7}{5}, y = \frac{5}{5}x=57,y=55
Let me know if you need any further explanations or have additional questions!
Also Read: Class 8 Mathematics Chapter 2 Linear Equations in One Variable
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As a seasoned content writer specialized in the fitness and health niche, Arun Bhagat has always wanted to promote wellness. After gaining proper certification as a gym trainer with in-depth knowledge of virtually all the information related to it, he exercised his flair for writing interesting, informative content to advise readers on their healthier lifestyle. His topics range from workout routines, nutrition, and mental health to strategies on how to be more fit in general. His writing is informative but inspiring for people to achieve their wellness goals as well. Arun is committed to equipping those he reaches with the insights and knowledge gained through fitness.