Solve the following question.x + |y| = 8, |x| + y = 6. How many pairs of x, y satisfy these two equations (in numerical value)?Correct answer is '1'. Can you explain this answer?

Question

Answer

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.

Answer

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.

Answer

Given Information:
The two equations are:
1. x * |y| = 8
2. |x| * y = 6

Solution:

To find the number of pairs of x and y that satisfy the given equations, we can start by analyzing each equation separately.

Equation 1:
x * |y| = 8

Since the absolute value of y is always positive, we can rewrite the equation as:
x * y = 8

Equation 2:
|x| * y = 6

This equation implies that the absolute value of x multiplied by y is equal to 6.

Case 1: x > 0 and y > 0
In this case, both x and y are positive. From equation 1, we know that x * y = 8. Since x and y are both positive, the only positive values that satisfy this equation are x = 8 and y = 1. However, when we substitute these values into equation 2, we find that the equation is not satisfied. Therefore, there are no solutions in this case.

Case 2: x > 0 and y < />
In this case, x is positive and y is negative. From equation 1, we know that x * (-y) = 8. Since x is positive and -y is also positive, the only positive values that satisfy this equation are x = 8 and y = -1. When we substitute these values into equation 2, we find that the equation is indeed satisfied. Therefore, there is one solution in this case.

Case 3: x < 0='' and='' y='' /> 0
In this case, x is negative and y is positive. From equation 1, we know that (-x) * y = 8. Since -x is positive and y is also positive, the only positive values that satisfy this equation are x = -8 and y = 1. When we substitute these values into equation 2, we find that the equation is indeed satisfied. Therefore, there is one solution in this case.

Case 4: x < 0='' and='' y='' />< />
In this case, both x and y are negative. From equation 1, we know that (-x) * (-y) = 8. Since -x and -y are both positive, the only positive values that satisfy this equation are x = -8 and y = -1. However, when we substitute these values into equation 2, we find that the equation is not satisfied. Therefore, there are no solutions in this case.

Conclusion:
After analyzing all possible cases, we find that there is only one pair of x and y that satisfy both equations, which is x = 8 and y = -1. Hence, the correct answer is '1'.

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