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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