Question

If 3x+2|y|+y=7 and x+|x|+3y=1, then x+2y is

• A. 0
• B. 1
• C. \(\frac{-4}{3}\)
• D. \(\frac{8}{3}\)

Answer 1

The Correct Option is A

Let's solve for x and y using the given equations: 

Given: 
1)\(3x + 2|y| + y = 7\)
2) \(x + |x| + 3y = 1\)

From the second equation: 
\(x + |x| = 1 - 3y\)

Case 1: x ≥ 0 In this case,\(|x| = x\), so: \(x + x = 1 - 3y\)    \(2x = 1 - 3y\)   \(x = 0.5 - 1.5y\)... (i) 
Case 2: x < 0 In this case, \(|x| = -x\), so: \(x - x = 1 - 3y\)

This gives us 0 = 1 - 3y, which is not possible. 

Hence, the first case is our valid scenario. 

Substitute the value of x from equation (i) into the first equation: 
\(3(0.5 - 1.5y) + 2|y| + y = 7\)

Expanding: 
\(1.5 - 4.5y + 2|y| + y = 7\)
\(1.5 - 3.5y + 2|y| = 7\)
\(-3.5y + 2|y| = 5.5\)

Now, for y: 

Case 1: y ≥ 0 

In this case, \(|y| = y\): 
\(-3.5y + 2y = 5.5\)
\(-1.5y = 5.5\)

This gives a negative value for y, which is not possible in this case. 

Case 2: y < 0 In this case, 
\(|y| = -y\): \(-3.5y - 2y = 5.5\)
\(-5.5y = 5.5\)
\(y = -1\)

Substituting this value of y in equation (i): 
\(x = 0.5 - 1.5(-1)\)
\(x = 0.5 + 1.5 = 2\)

So,\(x = 2\) and \(y = -1\). 

Finally, \(x + 2y = 2 + 2(-1) = 2 - 2 = 0\)

Answer 2

We must check for all regions: 
\(x≥0,y≥0\)
\(x≥0,y<0\)
However, once we find the answer for any one of these regions, we don't need to calculate for the others because the options suggest that there will be a single answer.

For \(x≥0,y≥0:\)
\(3x+3y=7\)
\(2x+3y=1\)
On solving,
\(x=6 \) and \(y =\)\(\frac {-11}{3}\)​.
Since \(y≥0\), this solution does not meet the given criteria.

For \(x≥0,y<0:\)
\( 3x−y=7\)
\(2x+3y=1\)
On solving, 
\(x=2\) and \(y=-1\).
This solution satisfies both conditions, so it is the correct point.
Now,
\(x+2y\)
\(= 2+2(−1)\)
\(=2−2\)
\(=0\)

So, the correct option is (A): \(0\)