If `|(3x,4),(5,x)|=|(4,-3),(5,-2)|`, then x =

Options

A. 3 only
B. `-3` only
C. 3 or `-3`
D. 6 or `-6`

Answer

To solve the equation \( |(3x, 4), (5, x)| = |(4, -3), (5, -2)| \), we will first calculate the determinants on both sides and then equate them.

### Step 1: Calculate the determinant on the left side
The determinant of a 2x2 matrix \( |(a, b), (c, d)| \) is given by the formula \( ad - bc \).

For the left side:
\[ |(3x, 4), (5, x)| = (3x)(x) - (4)(5) = 3x^2 - 20 \]

### Step 2: Calculate the determinant on the right side
For the right side:
\[ |(4, -3), (5, -2)| = (4)(-2) - (-3)(5) = -8 + 15 = 7 \]

### Step 3: Set the two determinants equal to each other
Now we equate the two results:
\[ 3x^2 - 20 = 7 \]

### Step 4: Solve for \( x \)
Add 20 to both sides:
\[ 3x^2 = 27 \]
Now, divide both sides by 3:
\[ x^2 = 9 \]
Finally, take the square root of both sides:
\[ x = \pm 3 \]

### Final Answer
Thus, the values of \( x \) are \( x = 3 \) and \( x = -3 \).

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