Problem 48 - College Algebra

Question

Solve each equation for \(x .\) $$\left|\begin{array}{rrr}2 & x & 1 \\\\-3 & 1 & 0 \\\2 & 1 & 4\end{array}\right|=39$$

Step-by-Step Solution

Answer

The solution for \(x\) is \(x=17/6\).

Step 1: Calculate the determinant of the 3x3 matrix

Start with the determinant formula for a 3x3 matrix: \(aei+bfg+cdh-ceg-bdi-afh\). In our case, a=2, b=x, c=1, d=-3, e=1, f=0, g=2, h=1 and i=4. After filling these values into the formula, the determinant will be: \(2(1*4+0*1)-x(-3*4+1*2)+1(-3*1+0*2)\)

Step 2: Simplify the equation

Simplify the above equation: 2(4)+x(12)+(-3). This simplifies to 8+12x-3

Step 3: Equate the determinant to 39 and solve for x

Now that we have the determinant equation, we equate it to 39 and solve for x. Hence 8+12x-3=39. This simplifies to 12x+5=39, giving 12x=34 and finally x=34/12 = 17/6.

Step 4: Check the solution

Substitute x=17/6 back into the determinant equation to verify if the left side equals 39. If true, then the solution is correct. Otherwise, recheck the steps.

Key Concepts

3x3 Matrix

A 3x3 matrix is a square array that consists of three rows and three columns. You can think of it like a small table with nine elements. In mathematical notation, it often looks like this:
\[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]
Each position in the matrix is defined by its row and column numbers. For instance, in our exercise, the 3x3 matrix is:
\[ \begin{bmatrix} 2 & x & 1 \ -3 & 1 & 0 \ 2 & 1 & 4 \end{bmatrix} \]
Here, the first number row (2, x, 1) is the matrix’s first row, and similarly, the other numbers follow suit in their respective positions. Understanding how to interpret and position these elements is crucial when dealing with more complex calculations such as determinants.

Matrix Equation

A matrix equation involves matrices in the expressions, similar to how variables are used in algebraic equations. Solving these equations requires different methods as compared to simple algebra equations.
In our example, the matrix equation was set by equating the determinant of the matrix to a number:
\[ |A| = 39 \]
Where \(|A|\) refers to the determinant of the matrix defined above. Discovering the value of a variable within a matrix equation often involves manipulating the matrix’s elements and using specific matrix-based techniques.

• Matrix equations still follow rules of algebra but are a step into more complex arithmetic.
• They can represent systems of equations compactly.
• They are powerful tools in linear algebra for solving problems involving multiple simultaneous equations.

Solving Equations

Solving equations in algebra means finding the value of the variable that makes the equation true. For our exercise, this involves determining the value of \(x\) that satisfies the matrix equation.
The process involves:

• Substituting the known values into the determinant formula for the given 3x3 matrix.
• Simplifying this expression by performing standard arithmetic operations: addition, subtraction, and multiplication.
• Setting the simplified expression equal to a specific value (in this case 39) and solving for \(x\).

By working through these steps carefully, you can isolate the variable \(x\) and determine its value. Ensuring your operations are accurate is key to finding the correct solution.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to solve. It's a fundamental skill in solving matrix equations and involves operations like expanding brackets, combining like terms, and isolating variables.
In the given exercise, by algebraically manipulating the expression obtained from the determinant computation, we can find the value of \(x\). Here’s how it unfolds:

• You start by expanding the expressions from the determinant formula and simplifying the resulting terms.
• Combine like terms, such as constants and the terms involving \(x\).
• Use basic arithmetic operations to isolate \(x\), turning the equation into a form like \(12x = 34\).
• Finally, solve for \(x\) by dividing both sides by the coefficient of \(x\) (here it is 12), resulting in \(x = \frac{34}{12} = \frac{17}{6}\).

Each step requires careful consideration of arithmetic rules and ensures that we solve for the correct value without making mistakes.