Consider the system of equations in variables
𝑥
,
𝑦

−
2
𝑥
+
(
𝑎
−
1
)
𝑦

=
0

(
𝑎
+
2
)
𝑥
−
2
𝑦

=
0

For how many values of
𝑎
does the system have infinitely many solutions?

Question

Consider the system of equations in variables 
𝑥
,
𝑦

−
2
𝑥
+
(
𝑎
−
1
)
𝑦
	
=
0

(
𝑎
+
2
)
𝑥
−
2
𝑦
	
=
0

For how many values of 
𝑎
 does the system have infinitely many solutions?

BlackTom AI · Accepted Answer

We start by restating the problem to focus on when the 2×2 linear system in x and y has infinitely many solutions.
The system is:
1) -2x + (a - 1) y = 0
2) (a + 2) x - 2y = 0
To have infinitely many solutions in a homogeneous linear system, the equations must be dependent, i.e., the two equations represent the same line. This occurs precisely when the determinant of the coefficient matrix is zero.
The coefficient matrix is [[-2, a-1], [a+2, -2]]. Its determinant is:
det = (-2)(-2) - (a-1)(a+2) = 4 - (a-1)(a+2).
Compute (a-1)(a+2) = a^2 + a - 2, so det = 4 - (a^2 + a - 2) = 6 - a^2 - a = -(a^2 + a - 6).
Set det = 0 to find potential a values:
-(a^2 + a - 6) = 0  ->  a^2 + a - 6 = 0.
Factor: (a + 3)(a - 2) = 0, so a = -3 or a = 2.
Now verify that for these a-values the equations are multiples (consistent) rather than contradictory.
Case a = 2:
Equation 1 becomes -2x + (2 - 1)y = -2x + y = 0.
Equation 2 becomes (2 + 2)x - 2y = 4x - 2y = 0, which simplifies to 2x - y = 0, i.e. y = 2x. The two equations are equivalent, so they describe the same line and yield infinitely many solutions along that line.
Case a = -3:
Equation 1 becomes -2x + (-3 - 1)y = -2x - 4y = 0, which reduces to x + 2y = 0.
Equation 2 becomes (-3 + 2)x - 2y = (-1)x - 2y = 0, which also reduces to x + 2y = 0. Again, the equations coincide, giving infinitely many solutions along x = -2y.
For any other a, the determinant is nonzero, meaning the two equations intersect at exactly one point (the trivial case x = y = 0 for a homogeneous system is unique only when the determinant is nonzero), so there would be a single solution rather than infinitely many.
Therefore, the system has infinitely many solutions for exactly two values of a: a = -3 and a = 2.
Among the provided options, the count of such a values is 2, which aligns with our analysis.

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