Questions 1-8 : Solve each system by elimination.
Question 1 :
x + 2y = 7
x – 2y = 1
Question 2 :
3x + y = 8
5x + y = 10
Question 3 :
x + ʸ⁄₂ = 4
ˣ⁄₃ + 2y = 5
Question 4 :
11x - 7y = xy
9x - 4y = 6xy
Question 5 :
³⁄_{y} + ⁵⁄ₓ = ²⁰⁄ₓ_{y}
2⁄ₓ + ⁵⁄_{y} = 15⁄ₓ_{y}
Question 6 :
8x – 3y = 5xy
6x – 5y = -2xy
Question 7 :
13x + 11y = 70
11x + 13y = 74
Question 8 :
65x – 33y = 97
33x – 65y = 1
1. Answer :
x + 2y = 7 ----(1)
x – 2y = 1 ----(2)
(1) + (2) :
Divide both sides by 2.
x = 4
Substitute x = 8 into (1).
4 + 2y = 7
Subtract 4 from both sides.
2y = 3
Divide both sides by 2.
y = ³⁄₂
Thereforem the solution is
(x, y) = (4, ³⁄₂)
2. Answer :
3x + y = 8 ----(1)
5x + y = 10 ----(2)
(2) - (1) :
Divide both sides by 2.
x = 4
Substitute x = 8 into (1).
4 + 2y = 7
Subtract 4 from both sides.
2y = 3
Divide both sides by 2.
y = ³⁄₂
Thereforem the solution is
(x, y) = (4, ³⁄₂)
3. Answer :
x + ʸ⁄₂ = 4 ----(1)
ˣ⁄₃ + 2y = 5 ----(2)
Multiply (1) by 2.
2x + y = 8 ----(3)
Multiply (2) by 3.
x + 6y = 15 ----(4)
2(4) - (3) :
Divide both sides by 11.
y = 2
Substitute y = 2 into (4).
x + 6(2) = 15
x + 12 = 15
Subtract 12 from both sides.
x = 3
Thereforem the solution is
(x, y) = (3, 2)
4. Answer :
11x - 7y = xy ----(1)
9x - 4y = 6xy ----(2)
Divide both sides of (1) by xy.
¹¹⁄y - ⁷⁄ₓ = 1
9⁄y - 4⁄ₓ = 6
Let a = 1⁄ₓ and b = 1⁄y.
Then, we have
11b + 7a = 1 ----(3)
9b - 4a = 6 ----(4)
9(3) - 11(4) :
Divide both sides by -19.
a = 3
Substitute a = 3 into (4).
9b - 4(3) = 6
9b - 12 = 6
9b = 18
Divide both sides by 2.
b = 2
a = 3 1⁄ₓ = 3 x = ⅓ |
b = 2 1⁄y = 2 y = ½ |
Therefore, the solution is
(x, y) = (⅓, ½)
5. Answer :
³⁄_{y} + ⁵⁄ₓ = ²⁰⁄ₓ_{y} ----(1)
2⁄ₓ + ⁵⁄_{y} = 15⁄ₓ_{y} ----(2)
Multiply both sides of (1) by xy.
3x + 5y = 20 ----(3)
Multiply both sides of (2) by xy.
2x + 5y = 15 ----(4)
(3) - (4) :
(3x + 5y) - (2x + 5y) = 20 - 15
3x + 5y - 2x - 5y = 5
x = 5
Substitute x = 5 into (4).
2(5) + 5y = 15
10 + 5y = 15
Subtract 10 from both sides.
5y = 5
Divide both sides by 5.
y = 1
Therefore, the solution is
(x, y) = (5, 1)
6. Answer :
8x – 3y = 5xy ----(1)
6x – 5y = -2xy ----(2)
Divide both sides of (1) by xy.
8⁄y - 3⁄ₓ = 5
6⁄y - 5⁄ₓ = -2
Let a = 1⁄ₓ and b = 1⁄y.
Then, we have
8b - 3a = 5 ----(3)
6b - 5a = -2 ----(4)
5(3) - 3(4) :
5(8b - 3a) - 3(6b - 5a) = 5(5) - 3(-2)
40b - 15a - 18b + 15a = 25 + 6
22b = 31
Divide botyh sides by 22.
b = ³¹⁄₂₂
Substitute b = ³¹⁄₂₂ into (4).
8(³¹⁄₂₂) - 3a = 5
¹²⁴⁄₁₁ - 3a = 5
Multiply both sides by 11.
124 - 33a = 55
Subtract 124 from both sides.
-33a = -69
Divide both sides by -33.
a = ²³⁄₁₁
a = ²³⁄₁₁ 1⁄ₓ = ²³⁄₁₁ x = ¹¹⁄₂₃ |
b = ³¹⁄₂₂ 1⁄y = ³¹⁄₂₂ y = ²²⁄₃₁ |
Therefore, the solution is
(x, y) = (¹¹⁄₂₃, ²²⁄₃₁)
7. Answer :
13x + 11y = 70 ----(1)
11x + 13y = 74 ----(2)
coefficient of x in (1) = coefficient of y in (2)
coefficient of y in (1) = coefficient of x in (2)
(1) + (2) :
24x + 24y = 144
Divide both sides by 24.
x + y = 6 ----(3)
(1) - (2) :
2x – 2y = -4
Divide both sides by 2.
x – y = -2 ----(4)
(3) + (4) :
2x = 4
x = 2
Substitute x = 2 into (3).
2 + y = 6
y = 4
Therefore, the solution is
(x, y) = (2, 4)
8. Answer :
65x – 33y = 97 ----(1)
33x – 65y = 1 ----(2)
coefficient of x in (1) = coefficient of y in (2)
coefficient of y in (1) = coefficient of x in (2)
(1) + (2) :
98x - 98y = 98
Divide both sides by 98.
x - y = 1 ----(3)
(1) - (2) :
32x + 32y = 96
Divide both sides by 32.
x + y = 3 ----(4)
(3) + (4) :
2x = 4
x = 2
Substitute x = 2 into (4).
2 + y = 3
y = 1
Therefore, the solution is
(x, y) = (2, 1)
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