Let us consider, c to be chickens and d to be ducks,
Given, 50c + 30d = 550,
Dividing, both sides with 10, we get
5 + 3 = 55 (Equation 1)
44 + 36 = 532
Dividing both sides with 4, we get
11 + 9 = 133 (Equation 2),
Multiplying Equation 1 by 3 we need to eliminate the y term, when subtracting Equation 1 from Equation 2,
Then, this equation becomes,
15 + 9 = 165 (Equation 3),
Now, subtracting equation 2 from equation 3, we get
(15 – 11 ) = ( 165 – 133 )
4 = 32,
Or, c = 8
Now, by substituting value of c in Equation 1, we get,
( 5* 8 ) + 3 = 55
Or, 3 = 55- 40 = 15
Or , d = 15/3 = 5
The substitution method is really helpful in case of systems for 2 equations, are two unknown factors. The prime idea is to focus on solving one of the equations, for one of the unknown factor, then substitute the result into the other equation.
Substitution method can be applied in four steps,
Step 1 involves solving the equation for either of the value od x = or y =, then the substitution is done for the solution from step 1 to the other equation, then the next stop involves solving the new equation and final step involves solving the second variable.
Example: Solve the following sum using substitution.
2 x + 3 y = 5
X + y = 5
Step 1: Solve one of the equations for either the condition x = or for the condition y = . The solution for second equation will be done for y,
x + y = 5
or , y = 5 – x
Step 2: Substitute the solution given in step 1, to the second equation
2 x + 3 y = 5
Or, 2 x + 3 ( 5- x ) = 5
Step 3: Solve this provided new equation.
2 x + 3 ( 5 – x ) = 5
Or, x + 15 – 3 x = 5
Or, - x + 15 = 5
Or, -x + 15 = 5
Or, - x = 5 – 15
Or, x = 10
Step 4: Solve the equation for the second variable.
Y = 5 – x
The solution is : ( x , y ) = (10, -5)
Ote: This must be kept in mind that the solution is same irrespective of the fact whether, first and which is the second. This is always more convenient to choose the one which is easier.
Example 2: Solve the equation by substitution.
2 x + 5 y = 12
4 x – y = 2
Step 1: Solve one of the equations for either x = or y = .Since, the co-efficient of y in equation 2 is -1, so it is comparatively easier to solve for y in equation 2,
4 x – y = 2
Or, -y = 2 – 4 x
Or, y = 4 x – 2
Step 2: Substituting the solution from step 1 to the second equation,
2 x = 5 y = 12
Or, 2 x + 5 ( 4 x – 2 ) = 12
Step 3: Solve the new equation, for the value of x,
2 x + 5 ( 4x – 2 ) = 12
Or, 2 x + 2 x + 20 x – 10 = 12
Or, 22 x = 22
Or, x = 1
Step 4: Solve for the second variable,
y = 4 x – 2
or, y = 4 . x – 2
or, y = 2.
Thus for the given equation of the price of ducks and chickens, the solution can be provided in the form of,
44 c + 36 d = 532
Or, 11 c + 9 d = 133
Or, 9 d = - 11 c + 133
Or, d = ( -11 c + 133) / 9
Or, 50 c + 30 d = 550
Or, 5 c + 3 d = 55
Or, 5 c + 3 [( - 11 c + 133)/ 9] = 55
Or, 5 c + [( - 11 c + 133 )/ 3] = 55
Or, ( 15c – 11c + 133) / 3 = 165
Or, 4 c + 133 = 165
Or, 4 c = 32
Or, c = 8,
Now, by substituting the value of the chicken from the above equation we put that value in the next equation to solve and find the value to be 5.
For, substitution equation, it is solved by this system of equation,
3 x + y = - 3
Or, x = - y + 3
The second equation is solved for x, so that the value can be substituted from the expression, -y + 3 in for x in the first equation,
3 x + y = - 3
Or, 3(-y + 3) + y = -3
Or, -3y + 9 + y = -3
Or, -3y + 9 + y = -3
0r, -2 y = -12
Or, y = 6,
Putting this value back in one of the original equations, where x = - y + 3 , we solve for the other variable,
x = - y + 3
or, x = -(6) + 3
or, x = -3,
Thus, the solution to the given equation is, x = - 3, y = 6
The same sum can be checked by plugging in the value to the original equation from these number, 3 x + y = -3
Or, 3 x + y = -3
Or, 3 ( -3 ) + 6 = -3,
Or, -9 + 6 = -3,
Or, -3 = -3
Or, y = 5 – 10
Or, y = - 5.
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