Solve the following simultaneous equations.
(1) 2x − 3y = 15 ; 8x + 5y = 77
Step 1 : Assume
Let 1x = m and 1y = n.
2m − 3n = 15 ...(1)
8m + 5n = 77 ...(2)
Step 2 : Solve the equations
Multiply (1) by 5 and (2) by 3:
10m − 15n = 75
24m + 15n = 231
Add:
10m − 15n = 75
24m + 15n = 231
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34m = 306
m = 9
Step 3 : Find n
Put m = 9 in (1):
2(9) − 3n = 15
18 − 3n = 15
−3n = −3
n = 1
Step 4 : Replace m and n
1x = 9 ⇒ x = 19
1y = 1 ⇒ y = 1
Hence, the solution is x = 19, y = 1.
(2) 10x + y + 2x − y = 4 ; 15x + y − 5x − y = −2
Step 1 : Assume
Let 1(x+y) = m and 1/(x−y) = n.
10m + 2n = 4 ...(1)
15m − 5n = −2 ...(2)
Step 2 : Solve the equations
Multiply (1) by 5 and (2) by 2:
50m + 10n = 20
30m − 10n = −4
Add:
50m + 10n = 20
30m − 10n = −4
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80m = 16
m = 15
Put m = 15 in (1): 2 + 2n = 4 ⇒ n = 1
Step 3 : Replace m and n
1(x+y) = 15 ⇒ x + y = 5 ...(3)
1/(x−y) = 1 ⇒ x − y = 1 ...(4)
Add (3) and (4):
x + y = 5
x − y = 1
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2x = 6
x = 3
Subtract (4) from (3):
x + y = 5
− (x − y = 1)
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2y = 4
y = 2
Hence, the solution is x = 3, y = 2.
(3) 27x − 2 + 31y + 3 = 85 ; 31x − 2 + 27y + 3 = 89
Step 1 : Assume
Let 1/(x−2) = m and 1(y+3) = n.
27m + 31n = 85 ...(1)
31m + 27n = 89 ...(2)
Step 2 : Add the equations
27m + 31n = 85
31m + 27n = 89
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58m + 58n = 174
m + n = 3 ...(3)
Step 3 : Subtract the equations
27m + 31n = 85
− (31m + 27n = 89)
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−4m + 4n = −4
m − n = 1 ...(4)
Step 4 : Find m and n
Add (3) and (4):
m + n = 3
m − n = 1
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2m = 4
m = 2
Subtract:
m + n = 3
− (m − n = 1)
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2n = 2
n = 1
Step 5 : Replace m and n
1/(x−2) = 2 ⇒ x−2 = 12 ⇒ x = 52
1(y+3) = 1 ⇒ y+3 = 1 ⇒ y = −2
Hence, the solution is x = 52, y = −2.
(4) 13x + y + 13x − y = 34 ; 12(3x + y) − 12(3x − y) = 18
Step 1 : Assume
Let 1(3x+y) = m and 1/(3x−y) = n.
m + n = 34 ...(1)
(12)m − (12)n = 18 ...(2)
Step 2 : Simplify the equations
Multiply (2) by 2:
m − n = 14 ...(3)
Step 3 : Add and subtract
Add (1) and (3):
m + n = 34
m − n = 14
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2m = 1
m = 12
Subtract (3) from (1):
m + n = 34
− (m − n = 14)
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2n = 12
n = 14
Step 4 : Replace m and n
1(3x+y) = 12 ⇒ 3x + y = 2 ...(4)
1/(3x−y) = 14 ⇒ 3x − y = 4 ...(5)
Add (4) and (5):
3x + y = 2
3x − y = 4
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6x = 6
x = 1
Subtract (4) from (5):
3x − y = 4
− (3x + y = 2)
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−2y = 2
y = −1
Hence, the solution is x = 1, y = −1.