Saturday, 19 July 2008

Substitution | Simultaneous Equations

Substitution method is one of the method to solve simultaneous equations.

Elimination method is the other techniquqe.

But, do note, substitution method is not as straight forward as the Elimination method.

Concept of Substitution Method:
We replace or substitute one unknown, converted from another equation, to make only a single unknown appears in the equation. Read further to understand.....

Example of Solution:
X + 2y = 5 ----- (A)
2X + y = 7 ------ (B)

Step 1: Convert equation (A) from x + 2y = 5 to x = 5 - 2y
Step 2: Substitute the x = 5 - 2y into equation (B) ==> 2 (5 - 2y) + y = 7

This gives us a final expression of 10 - 4 y + y = 7 which became -3y = 7 - 10 = -3

This results in y = -3 / -3 = 1 ===> y = 1 (Answer) !

Step 3: Replace y = 1 back into any of the original equations (A) or (B), either one will do.

Let's select equation (A) for computing.

x + 2y = 5 ===> x + 2 (1) = 5 ===> x = 5 - 2 (1) = 5 - 2 = 3 ==> x = 3 (Answer)

We have found the two unknowns x = 3 and y = 1! Eureka!

NOTE:
Substitution method is best used when the set of equations are of different order or when the unknowns are in product (multiplied) form.

Elimination method will not work in this case!

See another example below (different order).

Example:
xy = 4 -------(C)
x + y = 5 ------(D)

Step 1: Convert one of the equation (any one) to have unknown as subject. We select equation (D) and unknown x as subject.

x + y = 5 ===> x = 5 - y -----(E)

Step 2: Substitute the new X of equation (E) into equation (C)
xy = 4 ===> (5 - y) y = 4 ===> y^2 - 5y + 4 = 0 (a quadratic equation)

Step 3: Solving the quadratic equation , in this instance, using factorisation method,
yields (y - 4) (y - 1) = 0.

Therefore, from above (y - 4) = 0 or (y - 1) = 0.
Answer: y = 4 or y = 1

Step 4: Replace answer for y into any of original equations

Select equation (D):
When y = 4, ==> x + y = 5 ==> x + 4 = 5 ===> x = 1 (Answer)
When y = 1, ==> x + y = 5 ==> x + 1 = 5 ===> x = 4 (Answer)

From the example shown, it needs more indirect steps compared to the simpler direct Elimination method. However, if one understand the concept of substitution to reduce the number of unknowns to one, then it will simpler.

To learn more methods to solve simultaneous equations, click this here for post.
:-)