Problem 30 - Algebra 2

Question

Solve for \(x\) if \(\operatorname{det}\left[\begin{array}{cc}{2} & {x} \\ {5} & {-3}\end{array}\right]=24\)

Step-by-Step Solution

Answer

The value of \(x\) is \(-6\).

Step 1: Understand the Determinant Formula

For a 2x2 matrix \(\left[\begin{array}{cc}{a} & {b} \ {c} & {d}\end{array}\right]\), the determinant is calculated as \(ad - bc\). We are given the matrix \(\left[\begin{array}{cc}{2} & {x} \ {5} & {-3}\end{array}\right]\), so \(a = 2\), \(b = x\), \(c = 5\), and \(d = -3\).

Step 2: Set Up the Determinant Equation

Substitute the values from the matrix into the determinant formula: \(2 \cdot (-3) - x \cdot 5 = 24\). This simplifies to \(-6 - 5x = 24\).

Step 3: Rearrange the Equation

We want to isolate \(x\), so first add 6 to both sides of the equation: \(-5x = 24 + 6\). This simplifies to \(-5x = 30\).

Step 4: Solve for x

To find \(x\), divide both sides of the equation by -5 to get \(x = \frac{30}{-5}\), which simplifies to \(x = -6\).

Key Concepts

Solving Equations

When faced with solving equations, we aim to find the value of the unknown variable. In this exercise, the variable is represented by \(x\).

To begin solving the equation, we first focus on isolating the variable. This often involves performing the reverse of the operations applied to the variable.

For example:

• If the variable is multiplied by a number, we perform division to undo it.
• If a number is added to the variable, we subtract to nullify that effect.

In our problem, the equation \(-5x = 30\) shows that \(x\) is multiplied by \(-5\). Hence, to isolate \(x\), we divide both sides of the equation by \(-5\), leading to the solution \(x = -6\). This step-by-step isolation is key to ensuring accuracy when solving mathematical equations.

2x2 Matrix

Matrices are arrays of numbers, symbols, or expressions, arranged in rows and columns. The simplest form of a matrix is the 2x2 matrix, which has 2 rows and 2 columns. In a 2x2 matrix like \(\left[\begin{array}{cc}{a} & {b} \ {c} & {d}\end{array}\right]\), each element has a specific position, identified by its row and column numbers.

To understand a 2x2 matrix better:

• \(a\) is in the first row, first column.
• \(b\) is in the first row, second column.
• \(c\) is in the second row, first column.
• \(d\) is in the second row, second column.

This matrix structure becomes essential when calculating the determinant or performing operations like addition, subtraction, and multiplication.

Matrices in Algebra

Matrices in algebra play an essential role in solving system equations. They simplify expressions and support operations that deal with varying amounts of data. Specifically, matrices are crucial in linear algebra as they help in representing linear transformations.

One significant operation we can perform on a matrix is finding its determinant. For a 2x2 matrix, as demonstrated in our original step-by-step solution, the determinant is computed using the formula \(ad-bc\).

Knowing how to write and interpret determinants allows us to solve equations involving unknown variables, such as solving for \(x\) in mathematical problems. Beyond that, matrices are widely used in different applications—ranging from computer graphics, optimization problems, to systems of linear equations—because they provide a clear and structured way to handle linear transformations and data.