If `|(2x,5),(8,x)|=|(6,-2),(7,3)|` then the value of `x` is

Options

A. `3`
B. `+-3`
C. `+-6`
D. `6`

Explanation

To solve the equation \( |(2x, 5), (8, x)| = |(6, -2), (7, 3)| \), we will first calculate the determinants on both sides and then solve for \( x \).

### Step 1: Calculate the determinant on the left-hand side

The determinant of a 2x2 matrix \( |(a, b), (c, d)| \) is calculated using the formula:

\[ ad - bc \]

For the matrix \( |(2x, 5), (8, x)| \):

- \( a = 2x \)
- \( b = 5 \)
- \( c = 8 \)
- \( d = x \)

Thus, the determinant is:

\[ |(2x, 5), (8, x)| = (2x)(x) - (5)(8) = 2x^2 - 40 \]

### Step 2: Calculate the determinant on the right-hand side

Now, we calculate the determinant of the matrix \( |(6, -2), (7, 3)| \):

- \( a = 6 \)
- \( b = -2 \)
- \( c = 7 \)
- \( d = 3 \)

The determinant is:

\[ |(6, -2), (7, 3)| = (6)(3) - (-2)(7) = 18 + 14 = 32 \]

### Step 3: Set the two determinants equal to each other

Now we set the left-hand side equal to the right-hand side:

\[ 2x^2 - 40 = 32 \]

### Step 4: Solve for \( x \)

To solve for \( x \), first add 40 to both sides:

\[ 2x^2 = 32 + 40 \]
\[ 2x^2 = 72 \]

Now, divide both sides by 2:

\[ x^2 = \frac{72}{2} = 36 \]

Next, take the square root of both sides:

\[ x = \pm 6 \]

### Final Answer

Thus, the values of \( x \) are:

\[ x = 6 \quad \text{or} \quad x = -6 \]

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