Problem 26 - A Graphical Approach to College Algebra

Question

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rr}5 & x \\\\-3 & 2\end{array}\right]=6$$

Step-by-Step Solution

Answer

x = -\frac{4}{3}.

Step 1: Write down the determinant expression

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). Apply the formula to \( \begin{bmatrix} 5 & x \ -3 & 2 \end{bmatrix} \) using \( a = 5, b = x, c = -3, d = 2 \).

Step 2: Set up the equation for the determinant

Using the determinant formula, the equation becomes \( 5 \cdot 2 - (-3) \cdot x = 6 \). Simplify it to get \( 10 + 3x = 6 \).

Step 3: Solve for x

Re-arrange the equation \( 10 + 3x = 6 \) to solve for \( x \). Subtract 10 from both sides to get \( 3x = -4 \). Then, divide both sides by 3 to find \( x = -\frac{4}{3} \).

Key Concepts

2x2 Matrices

Let's delve into the heart of a 2x2 matrix before we solve problems involving them. A 2x2 matrix is essentially a grid containing four numbers arranged in two rows and two columns. Consider the matrix: \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]Here, "a," "b," "c," and "d" are elements of the matrix. Each position within the matrix holds a specific role when performing mathematical operations, like finding its determinant.

• The "a" and "d" are the elements of the main diagonal when you draw from top left to bottom right.
• The "b" and "c," positioned off this diagonal, are just as crucial as they counterbalance the diagonal elements in determining specific properties of matrices.

This simple square matrix format is foundational in linear algebra, where we can quickly compute determinants and facilitate the transition into solving matrix equations.

Solving Equations

Solving equations, especially those involving matrices, involves methodically breaking down the given expressions to isolate the unknown variable. In our case, it begins as:\[5 \cdot 2 - (-3) \cdot x = 6\]Notice here that our equation originates from the determinant of the matrix.

• First, we compute the product of the main diagonal \( a \cdot d \), yielding \( 10 \).
• The off-diagonal product \( b \cdot c \) results in \( +3x \) as multiplication of negative values changes the sign.

Finding \( x \) means rearranging, which initially demands simplifying to \( 10 + 3x = 6 \). After simplification, subtract 10 from both sides, simplifying further to:\[3x = -4\]Finally, divide each side by 3, solving for \( x \), yielding \( x = -\frac{4}{3} \). Methodical patience is essential here, ensuring each mathematical step is concrete and accurate.

Algebraic Expressions

When dealing with algebraic expressions, we manipulate symbols following algebraic rules to solve for unknowns.

• Our goal is often to simplify expressions, making it easier to solve equations and find roots or other elements of interest.
• Understand that each variable, constant, and operator serves specific functions.

Beginning with our equation, simplification is key:Starting from \[10 + 3x = 6\]We reorganize by subtracting 10,\[3x = -4\]and subsequently, we divide the expression by 3 to obtain a concise answer for \( x \); transforming sometimes messy looking equations into manageable parts. Algebraic skills, like manipulating expressions and solving linear equations, are foundational and enable us to tackle complex problems with confidence. They ensure you have the toolkit necessary to confront and solve equations stemming from real-world queries or academic challenges.