Algebra

To solve a system of equations involving two variables, one knows from high school that two equations are required. This rule needs to be corrected. The correct rule is you can solve a system of equations involving two variables if and only if you have two distinct linear equations.

Applying this rule incorrectly, particularly in Data sufficiency type questions, increases the chance of error.

For Example:

Q. What is the value of x?

1) x – 4y/7 = 8

2) 7x = 4y + 56

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Approach:

If one solves this question hastily, keeping the above rule in mind, he will choose (C) immediately. However, the correct answer is (E).

Solution:

Statement # 1 simplification:

x – 4y/7 = 8

7x – 4y = 56

Statement 1 is insufficient because one cannot solve a linear equation involving two variables with one equation.

Statement # 2 simplification:

7x = 4y + 56

7x – 4y = 56

This statement is also insufficient.

After simplification, statement 1 and statement 2 are effectively the same equations; this means that one cannot solve these two equations by using any method.

Example #2:

What is the value of x?

1) x – 6/y = 4

2) 2x + y = 16

Statements 1 and 2 are insufficient because of two variables in a single equation.

Now let’s solve taking both statements together:

Begin with the first equation.

x – 6/y = 4

Isolate x.

x = 6/y + 4

Substitute the value of x in the second equation

2(6/y + 4) +y = 16

After applying the distributive property of multiplication over parenthesis,

12/y + 8 + y =16

Multiply both sides of the equation by y to clear the fraction.

12 + 8y + y2 = 16y

Transfer all terms to one side of the equation

y2 – 8y + 12 = 0

This is a quadratic equation with two solutions, but in Data sufficiency, we need one unique answer for x, which is why both statements are insufficient.

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