Solve the following linear equations.
Question 1. x/2 - 1/5 = x/3 + 1/4
Solution:
(5x - 2)/10 = (4x + 3)/12 ...(Taking LCM on both the sides)
12(5x - 2) = 10 (4x + 3) ...(Cross multiplying)
60x - 24 = 40x + 30 ...(Solving the brackets)
60x - 40x = 30 + 24 ...(Transposing terms of x to LHS and others to RHS)
20x = 54
x = 54/20 or 27/10 ... (Solution)
Verification:
Putting value of "x" in the equation to check if our answer is correct
27/20 - 1/5 = 27/30 + 1/4
(27 - 4)/20 = (108 + 30)/120
23/20 = 138/120
23/20 = 23/20
LHS = RHS (Hence Proved that solution is correct)
Question 2. n/2 - 3n/4 + 5n/6 = 21
Solution:
(6n - 9n + 10n)/12 = 21 ...(Taking LCM and solving LHS)
7n/12 = 21 (Solving LHS)
7n = 21 × 12
n = 36 ...(Solution)
Verification:
Putting value of "n" in the equation to check if our answer is correct
36/2 - 108/4 + 180/6 = 21
18 - 27 + 30 = 21
21 = 21
LHS = RHS (Hence Proved that solution is correct)
Question 3. x + 7 - 8x/3 = 17/6 - 5x/2
Solution:
x - 8x/3 + 5x/2 = 17/6 - 7 ...(Transposing terms of x to LHS and others to RHS)
(6x - 16x + 15x)/6 = (17 - 42)/6 ...(Taking LCM and solving)
5x/6 = -25/6
x = -5 ...(Solution)
Verification -
Putting value of "x" in the equation to check if our answer is correct
-5 + 7 - (-40)/3 = 17/6 - (-25)/2
2 + 40/3 = 17/6 + 25/2
46/3 = (17 + 75)/6
46/3 = 92/6
46/3 = 46/3
LHS = RHS (Hence Proved that solution is correct)
Question 4. (x - 5)/3 = (x - 3)/5
Solution:
5(x - 5) = 3(x - 3) ...(Cross multiply)
5x - 25 = 3x - 9
2x = 16
x = 8 ...(Solution)
Verification -
Putting value of "x" in the equation to check if our answer is correct
(8 - 5)/3 = (8 - 3)/5
3/3 = 5/5
1 = 1
LHS = RHS (Hence Proved that solution is correct)
Question 5. (3t - 2)/4 - (2t + 3)/3 = 2/3 - t
Solution:
3t/4 - 1/2 - 2t/3 -1 = 2/3 - t ...(Solving brackets)
3t/4 - 2t/3 + t = 2/3 + 1 + 1/2 ...(Transposing terms of x to LHS and others to RHS)
(9t - 8t + 12t)/12 = (4 + 6 + 3)/6 ...(Taking LCM both sides)
13t/12 = 13/6
t = 2 ...(Solution)
Verification -
Putting value of "t" in the equation to check if our answer is correct
(3 × 2 - 2)/4 - (2 × 2 + 3)/3 = 2/3 - 2
4/4 - 7/3 = 2/3 - 2
(12 - 28)/12 = (2 - 6)/3
-16/12 = -4/3
-4/3 = -4/3
LHS = RHS (Hence Proved that solution is correct)
Question 6. m - (m - 1)/2 = 1 - (m - 2)/3
Solution:
(2m - m + 1)/2 = (3 - m + 2)/3 ...(Taking LCM both sides)
(m + 1)/2 = (5 - m)/3
3(m + 1) = 2(5 - m) ...(Cross multiplying)
3m + 3 = 10 - 2m
5m = 7
m = 7/5 ...(Solution)
Verification -
Putting value of "m" in the equation to check if our answer is correct
7/5 - (7/5 - 1)/2 = 1 - (7/5 - 2)/3
7/5 - 1/5 = 1 - (-3)/15
6/5 = 1 + 1/5
6/5 = 6/5
LHS = RHS (Hence Proved that solution is correct)
Question 7. 3(t - 3) = 5(2t + 1)
Solution:
3t - 9 = 10t + 5 ...(Opening brackets)
3t - 10t = 9 + 5
-7t = 14
t = -2 ...(Solution)
Verification -
Putting value of "t" in the equation to check if our answer is correct
3(-2 - 3) = 5(2(-2) + 1)
3(-5) = 5(-4 +1)
-15 = -15
LHS = RHS (Hence Proved that solution is correct)
Question 8. 15(y – 4) – 2(y – 9) + 5(y + 6) = 0
Solution:
15y - 60 - 2y + 18 + 5y + 30 = 0
18y - 12 = 0
y = 12/18 or 2/3 ...(Solution)
Verification -
Putting value of "y" in the equation to check if our answer is correct
15(2/3 - 4) - 2(2/3 - 9) + 5(2/3 + 6) = 0
10 - 60 - 4/3 +18 + 10/3 + 30 = 0
-50 -4/3 + 48 + 10/3 = 0
-2 + 6/3 = 0
-2 + 2 = 0
0 = 0
LHS = RHS (Hence Proved that solution is correct)
Question 9. 3(5z – 7) – 2(9z – 11) = 4(8z – 13) – 17
Solution:
15z - 21 - 18z + 22 = 32z - 52 - 17 ...(Solving the brackets)
-3z + 1 = 32z - 69
-35z = -70
z = 2 ...(Solution)
Verification -
Putting value of "z" in the equation to check if our answer is correct
3(5(2) - 7) - 2(9(2) - 11) = 4(8(2) - 13) - 17
3(3) - 2(7) = 4(3) - 17
9 - 14 = 12 - 17
-5 = -5
LHS = RHS (Hence Proved that solution is correct)
Question 10. 0.25(4f – 3) = 0.05(10f – 9)
Solution:
f - 0.25(3) = 0.5f - 0.05(9)
f - 0.75 = 0.5f - 0.45
0.5f = 0.75 - 0.45
f = 3/5 or 0.6 (Solution)
Verification -
Putting value of "f" in the equation to check if our answer is correct
0.25(4(0.6) - 3) = 0.05(10(0.6) - 9)
0.25(2.4 - 3) = 0.05(6 - 9)
0.25 × (-0.6) = 0.05 × (-3)
-0.15 = -0.15
LHS = RHS (Hence Proved that solution is correct)