If $\begin{vmatrix} 2 & -4 \\ x & 9 \end{vmatrix} = 58$, find the value of x.

Analysis

The question asks us to find the value of $x$ in a $2 \times 2$ determinant equation. To solve this, we must apply the standard formula for finding the determinant of a $2 \times 2$ matrix.

Concept: Determinant of a $2 \times 2$ Matrix

For any $2 \times 2$ matrix given as: $A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$

The determinant is calculated using the formula: $\text{det}(A) = (a \times d) - (b \times c)$

Step-by-Step Solution

Given the matrix: $\begin{vmatrix} 2 & -4 \\ x & 9 \end{vmatrix} = 58$

Step 1: Apply the determinant formula $(2 \times 9) - (-4 \times x) = 58$

Step 2: Simplify the multiplication $18 - (-4x) = 58$ $18 + 4x = 58$

Step 3: Isolate the term with $x$ Subtract 18 from both sides of the equation: $4x = 58 - 18$ $4x = 40$

Step 4: Solve for $x$ Divide both sides by 4: $x = \frac{40}{4}$ $x = 10$

Conclusion

Comparing our result to the provided options, the value of $x$ is 10.

Exam Tip

When dealing with determinants involving negative numbers, always be extra careful with the signs. Remember that subtracting a negative number (like $-(-4x)$) is the same as adding a positive number ($+4x$). This is a common place where students lose marks due to simple sign errors!