For which of the following values of a and b do the given equations have NO solution?

![Screenshot 2025-05-08 215059.png](https://res.cloudinary.com/aptidude/image/upload/v1746721268/aptidude_questions/uumjf5sstlnf4okdg6l7.png)

Question

For which of the following values of a and b do the given equations have NO solution?

![Screenshot 2025-05-08 215059.png](https://res.cloudinary.com/aptidude/image/upload/v1746721268/aptidude_questions/uumjf5sstlnf4okdg6l7.png)

Accepted Answer

Given the system of equations:
1) x - ay = 2 - a
2) (1 - a)x + 6y = a + b

We want to find values of a and b for which the system has NO solution.

Step 1: Write the system in standard form:
Equation 1: x - ay = 2 - a
Equation 2: (1 - a)x + 6y = a + b

Step 2: For a system of two linear equations:
Ax + By = C
Dx + Ey = F

The system has:
- Unique solution if (A/E) ≠ (D/B)
- Infinite solutions if (A/E) = (D/B) = (C/F)
- No solution if (A/E) = (D/B) ≠ (C/F)

Step 3: Identify coefficients:
A = 1, B = -a, C = 2 - a
D = 1 - a, E = 6, F = a + b

Step 4: Check the ratios:
A/E = 1/6
D/B = (1 - a)/(-a) = -(1 - a)/a

Set A/E = D/B for no solution or infinite solutions:
1/6 = -(1 - a)/a
Cross multiply:
a = -6(1 - a) = -6 + 6a
Bring terms together:
a - 6a = -6
-5a = -6
=> a = 6/5 = 1.2

But none of the options have a = 1.2, so check if the system can have no solution for other values.

Alternatively, check if the lines are parallel but not coincident.

Rewrite equations:
Equation 1: x - ay = 2 - a
=> x = ay + 2 - a

Equation 2: (1 - a)x + 6y = a + b
=> (1 - a)x = a + b - 6y
=> x = (a + b - 6y)/(1 - a), provided (1 - a) ≠ 0

For the lines to be parallel, their slopes must be equal.

Slope of equation 1:
From x - ay = 2 - a
=> x = ay + (2 - a)
Slope = coefficient of y in x = a

Slope of equation 2:
From (1 - a)x + 6y = a + b
=> (1 - a)x = a + b - 6y
=> x = (a + b)/(1 - a) - (6/(1 - a))y
Slope = -6/(1 - a)

Set slopes equal:
a = -6/(1 - a)
Cross multiply:
a(1 - a) = -6
=> a - a^2 = -6
=> a^2 - a - 6 = 0

Solve quadratic:
a^2 - a - 6 = 0
Discriminant = 1 + 24 = 25
Roots:
a = [1 ± 5]/2
=> a = (1 + 5)/2 = 6/2 = 3
or a = (1 - 5)/2 = -4/2 = -2

Check which a values are in options: a = 3 is present.

Step 5: For no solution, the lines must be parallel but not coincident.
Check if the constants satisfy the ratio condition:
For no solution:
C/E ≠ F/B
Calculate C/E and F/B for a = 3:
C = 2 - a = 2 - 3 = -1
E = 6
C/E = -1/6

F = a + b = 3 + b
B = -a = -3
F/B = (3 + b)/(-3) = -(3 + b)/3

For no solution:
C/E ≠ F/B
=> -1/6 ≠ -(3 + b)/3
Multiply both sides by 6*3=18:
-3 ≠ -6(3 + b)
-3 ≠ -18 - 6b
Add 18 both sides:
15 ≠ -6b
Divide by -6:
-15/6 ≠ b
=> b ≠ -2.5

But the options mention b ≠ -1 or b ≠ 1.

Check if b = -1 satisfies no solution:
b = -1
Check if C/E = F/B:
F/B = -(3 + (-1))/3 = -(2)/3 = -2/3
C/E = -1/6
-1/6 ≠ -2/3
So no solution condition holds for b ≠ -1.

Hence, for a = 3 and b ≠ -1, the system has no solution.

Therefore, the correct option is 2: a = 3, b ≠ -1.

Answer

Answer

Answer