Simultaneous Linear Equations
When two or more equations have the same variables, solving them together is called simultaneous equations.
Example: A customer pays Rs. 1050 for 3 kg oyster mushrooms and 2 kg white button mushrooms, while another pays Rs. 1200 for 2 kg oyster and 3 kg white button mushrooms.
Let cost of 1 kg oyster mushroom = x and 1 kg white button = y
Equations:
3x + 2y = 1050 …(i)
2x + 3y = 1200 …(ii)
Solving these equations gives the cost of each mushroom type.
Methods of Solving Simultaneous Equations
Substitution Method
Elimination Method
Graphical Method
Substitution Method
Example 1: Solve 3x + 2y = 1050 and 2x + 3y = 1200
Solution:
From equation (i):
3x + 2y = 1050
3x = 1050 - 2y
x = (1050 - 2y)/3
Substitute x in equation (ii):
2x + 3y = 1200
2*((1050 - 2y)/3) + 3y = 1200
(2100 - 4y)/3 + 3y = 1200
Multiply both sides by 3:
2100 - 4y + 9y = 3600
5y = 1500
y = 300
Substitute y in equation (i):
3x + 2*300 = 1050
3x + 600 = 1050
3x = 450
x = 150
Solution: Oyster mushroom = Rs.150/kg, White button mushroom = Rs.300/kg
Elimination Method
Example 2: Solve 4x + 3y = 100 and 5x + 2y = 90
Solution:
Multiply equation (i) by 2:
8x + 6y = 200
Multiply equation (ii) by 3:
15x + 6y = 270
Subtract the first from the second:
15x + 6y - (8x + 6y) = 270 - 200
7x = 70
x = 10
Substitute x in equation (i):
4*10 + 3y = 100
40 + 3y = 100
3y = 60
y = 20
Solution: Small copy = Rs.10, Large copy = Rs.20
Graphical Method
Express equations in slope-intercept form: y = mx + c
Plot points and draw lines
Intersection point = solution
Example 3: Solve 2x + 3y = 12 and x + 2y = 7
Solution:
y = (12 - 2x)/3
y = (7 - x)/2
Create table of values:
Equation 1: x=0 ⇒ y=4, x=3 ⇒ y=2, x=6 ⇒ y=0
Equation 2: x=1 ⇒ y=3, x=3 ⇒ y=2, x=5 ⇒ y=1
Intersection point = (3, 2)
Solution: x = 3, y = 2
Important Questions with Solutions
1. Solve x + y = 7 and x - y = 2
or, x = y + 2
or, (y + 2) + y = 7
or, 2y + 2 = 7
or, 2y = 5
or, y = 2.5
or, x = y + 2 = 4.5
2. Solve 3x - 2y = 11 and 3x + y = 15
or, From second equation: y = 15 - 3x
or, Substitute in first: 3x - 2(15 - 3x) = 11
or, 3x - 30 + 6x = 11
or, 9x = 41
or, x = 41/9
or, y = 15 - 3*(41/9) = 104/9
3. Solve 4x + 3y = 100 and 5x + 2y = 90
or, Use elimination method
or, x = 10, y = 20
4. Solve 3x + 2y = 11 and 4x - 3y = 9
or, Use elimination method
or, x = 3, y = 1
5. Solve 2x + 3y = 12 and x + 2y = 7
or, From second equation: x = 7 - 2y
or, Substitute in first: 2(7 - 2y) + 3y = 12
or, 14 - 4y + 3y = 12
or, -y + 14 = 12
or, y = 2
or, x = 7 - 2*2 = 3
6. Cost of chicken and mutton problem
or, 2x + y = 1500
or, x + 2y = 1950
or, Solve equations: x = 700, y = 1000
7. Rectangular playground: perimeter = 32, breadth = 1/3 length
or, 2(l + b) = 32 → l + b = 16
or, b = l/3 → l + l/3 = 16
or, 4l/3 = 16
or, l = 12
or, b = 4
8. Solve x + y = 16 and x - y = -4
or, x = y - 4
or, (y - 4) + y = 16
or, 2y - 4 = 16
or, 2y = 20
or, y = 10
or, x = y - 4 = 6
9. Solve 3x + y = 15 and 2x + 3y = 17
or, Multiply first by 3: 9x + 3y = 45
or, Subtract second equation: (9x + 3y) - (2x + 3y) = 45 - 17
or, 7x = 28
or, x = 4
or, Substitute x in first: 3*4 + y = 15 → y = 3
10. Solve 5x - y = 23 and 3x - 2y = 4
or, Multiply first by 2: 10x - 2y = 46
or, Subtract second: (10x - 2y) - (3x - 2y) = 46 - 4
or, 7x = 42
or, x = 6
or, Substitute x in second: 3*6 - 2y = 4 → 18 - 2y = 4
or, 2y = 14 → y = 7
11. Solve x + y = 25 and x - y = 5
or, x = y + 5
or, (y + 5) + y = 25 → 2y + 5 = 25
or, 2y = 20 → y = 10
or, x = y + 5 = 15
12. Solve x + 2y = 6 and 2x + y = 6
or, Multiply second by 2: 4x + 2y = 12
or, Subtract first: (4x + 2y) - (x + 2y) = 12 - 6
or, 3x = 6 → x = 2
or, Substitute x: 2 + 2y = 6 → 2y = 4 → y = 2
13. Solve 2x + y = 5 and x - y = 1
or, x = y + 1
or, 2(y + 1) + y = 5
or, 2y + 2 + y = 5 → 3y + 2 = 5
or, 3y = 3 → y = 1
or, x = y + 1 = 2
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