If x=1 , y=2 is a solution of the equation `a^(2)x+ay=3`, then find the values of a .

Options

A. a=1 , -3
B. a=1,2
C. a=-3,2
D. a=-1,-2

Answer

To solve the equation \( a^2 x + ay = 3 \) given that \( x = 1 \) and \( y = 2 \), we can follow these steps:

### Step 1: Substitute the values of x and y into the equation
We start with the equation:
\[ a^2 x + ay = 3 \]
Substituting \( x = 1 \) and \( y = 2 \):
\[ a^2(1) + a(2) = 3 \]

### Step 2: Simplify the equation
This simplifies to:
\[ a^2 + 2a = 3 \]

### Step 3: Rearrange the equation
We can rearrange this equation to set it to zero:
\[ a^2 + 2a - 3 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \(-3\) (the constant term) and add up to \(2\) (the coefficient of \(a\)). The numbers \(3\) and \(-1\) fit this requirement:
\[ (a + 3)(a - 1) = 0 \]

### Step 5: Solve for a
Now, we set each factor equal to zero:
1. \( a + 3 = 0 \) → \( a = -3 \)
2. \( a - 1 = 0 \) → \( a = 1 \)

### Conclusion
The values of \( a \) that satisfy the equation are:
\[ a = -3 \quad \text{or} \quad a = 1 \]

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