## Substitution method

Substitution method is another powerful technique of solving multiple choice questions besides elimination method.

Test takers must understand the use and power of substitution method. They should realize that in the actual test substitution method will pre-empt actual solving and thus save precious time.

Let us discuss substitution method in solving a few multiple choice questions

1. If │p – 8│ = 3p, then p is

A. – 4
B. – 2
C. 0
D. 1
E. 2

Before subscribing to the substitution method of solving the above multiple choice question, let us solve in the conventional way. It is
On removing the absolute value symbol ││

We know │x│ = ±x,

So, │p – 8 │ = ± (p – 8),

So, if │p – 8 │ = 3p, then

(p – 8) = 3p or – (p – 8) = 3p.

Let us find p, by solving the two equations separately.

If (p – 8) = 3p, then

p – 3p = 8, -2p = 8, so p = - 4,

And again,

– (p – 8) = 3p, p – 8 = -3p, so,

p + 3p = 8, i.e. 4p = 8, so, finally p = 2

Finally, p = 2 or p = - 4

But, it must be remembered that absolute values are never negative. Absolute values are only non-negative.

But – 4 for p will make the given absolute value equation equal to a negative number. This can be verified as follows.

On substitution of -4 in p in │p – 8 │ = 3p, in both LHS and RHS, we get

│- 4 – 8 │ = 3 (-4), i.e. │ 12 │ = –12.

But this is not correct, since

│ │ ≠ ve

But, on substitution of 2 in p in │p – 8 │ = 3p, in both LHS and RHS, we get

│- 2 – 8 │ = 3 (-2), i.e. │ 6│ = –6. And this is right.

So, p = 2 is the only acceptable value.

In spite of solving the given absolute value equation and obtaining two solutions for the variable p, it still remains to be checked if both the values of the variable are part of the solution set.

This is done by substitution of the values of p in the equation given in the question. And substitution shows that one value of p has to be discarded as it does not satisfy property of absolute values.

But this method of direct solving to find the solutions of the given absolute equation is quite time-consuming. And time is very precious in any test.

Also, direct solving is not the invariable way of solving the present question, which can as well be solved more easily and quickly using the substitution method.

This is how substitution method is more efficient:

The given absolute value equation is

│p – 8│ = 3p.

One by one substitute answer choice numbers and eliminate those values that are not acceptable.

Substitution of option A number – 4 in │p – 8│ = 3p, yields us

│–4 – 8│ = 3 (- 4), i.e. │–12│ = - 12, which is wrong (absolute values are never negative)

Option B number – 2 also is eliminated for the same reason as option A.

Substitution of Option C number 0 in │p – 8│ = 3p, gives

│0 – 8│ = 3 (0), i.e. │ – 8│ = 0, which is wrong too.

Substitution of option D number 1 in │p – 8│ = 3p, gives

│1 – 8│ = 3 (1), i.e. │– 7│ = 3, again not correct, and finally

│2 – 8│ = 3 (2), i.e. │– 6│ = 6, which is right.

This substitution method does not consume too much time like the direct method.

In solving multiple choice questions in which answers already appear, substitution method works very fast.

Never forget to use substitution method to solve multiple choice questions.

Use substitution method to solve the following multiple choice question

Which of the following answer choices contains the solution of the quadratic equation x2 – 5x + 6 = 0

A. x = 1
B. x = 2 and x = 3
C. x = 2 or x = 3
D. x = 5
E. x = 6.