Question 7

Explanation

We are given the following determinant equation:

The first step is to calculate the determinant of the matrices on the left and right sides. Recall that the determinant of a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $ad - bc$.

Determinant of the left side:

Determinant of the right side:

Now we equate both results:

Move $5$ to the left side to form a quadratic equation:

We factor this quadratic equation. We look for two numbers that multiply to $3 \times (-5) = -15$ and sum to $-2$. These numbers are $-5$ and $3$.

Thus, the solutions are:

So, the values of $x$ that satisfy the equation are $\frac{5}{3}$ and $-1$.