CAT Quantitative Aptitude Questions | CAT Algebra - Linear Equations; Quadratic Equations

The question is about Possible pairs of Solutions of a Linear Equation. We need to find out the possible pairs that satisfy the given linear equation which contains modulus. Framing and solving equations is an integral part of this topic. Get as much practice as you can in these two topics because the benefits of being good at framing equations will be useful in other concepts. Modulus of a number cannot be negative. A simple idea that somehow escapes our mind every now and then.

Question 3: x + |y| = 8, |x| + y = 6. How many pairs of x, y satisfy these two equations?

1. 2
2. 4
3. 0
4. 1

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Linear Equations: Modulus of a number cannot be negative. A simple idea that somehow escapes our mind every now and then.

We start with the knowledge that the modulus of a number can never be negative, though the number itself may be negative.

The first equation is a pair of lines defined by the equations
y = 8 – x ------- (i) (when y is positive)
y = x – 8 ------- (ii) (when y is negative)
With the condition that x ≤ 8 (because if x becomes more than 8, |y| will be forced to be negative, which is not allowed)

The second equation is a pair of lines defined by the equations:
y = 6 – x ------- (iii) (when x is positive)
y = 6 + x ------- (iv) (when x is negative)
with the condition that y cannot be greater than 6, because if y > 6, |x| will have to be negative.

On checking for the slopes, you will see that lines (i) and (iii) are parallel. Also (ii) and (iv) are parallel (same slope).
Lines (i) and (iv) will intersect, but only for x = 1; which is not possible as equation (iv) holds good only when x is negative.
Lines (ii) and (iii) do intersect within the given constraints. We get x = 7, y = -1. This satisfies both equations.
Only one solution is possible for this system of equations.

The question is "x + |y| = 8, |x| + y = 6. How many pairs of x, y satisfy these two equations?"

Hence the answer is "1"

Choice D is the correct answer.