Problem 38 - Algebra and Trigonometry

Question

Solve for \(x\) and \(y\) $$ \left[\begin{array}{rr}{x} & {y} \\ {-y} & {x}\end{array}\right]-\left[\begin{array}{rr}{y} & {x} \\ {x} & {-y}\end{array}\right]=\left[\begin{array}{rr}{4} & {-4} \\ {-6} & {6}\end{array}\right] $$

Step-by-Step Solution

Answer

The solution is \(x=5\) and \(y=1\).

Step 1: Simplify the Left-Hand Side

Subtract the matrices component-wise on the left-hand side. For the first element, \(x-y\), for the second \(y-x\), for the third \(-y-x\), and for the fourth \(x+y\). The equation becomes: \[\begin{pmatrix}x-y & y-x \-y-x & x+y\end{pmatrix} = \begin{pmatrix}4 & -4 \ -6 & 6 \end{pmatrix}\]

Step 2: Equate Corresponding Elements

Set the corresponding elements of the resulting matrices equal to each other. The equations are:1. \(x-y = 4\)2. \(y-x = -4\)3. \(-y-x = -6\)4. \(x+y = 6\).

Step 3: Solve Equation 1: \(x-y = 4\)

Solve for \(x\) from the equation \(x - y = 4\). Rearrange to find \(x = y + 4\).

Step 4: Solve Equation 4: \(x+y = 6\)

Utilize \(x = y + 4\) in substitution. Replace \(x\) in equation \(x + y = 6\) by \(y + 4\) to obtain \(y + 4 + y = 6\) which simplifies to \(2y + 4 = 6\).

Step 5: Calculate \(y\)

From the equation \(2y + 4 = 6\), simplify to \(2y = 2\) by subtracting 4 from both sides. Then, divide both sides by 2 to solve for \(y\), getting \(y = 1\).

Step 6: Calculate \(x\)

Substitute \(y = 1\) back into \(x = y + 4\). Thus, \(x = 1 + 4 = 5\).

Step 7: Verify Solutions

Plug \(x = 5\) and \(y = 1\) back into each initial equation to check correctness:- First element: \(x-y = 5-1 = 4\)- Second element: \(y-x = 1-5 = -4\)- Third element: \(-y-x = -1-5 = -6\)- Fourth element: \(x+y = 5+1 = 6\)All elements match the right side of the equation.

Key Concepts

Matrix Equations

Matrix equations are a compact way to represent and solve systems of linear equations. They utilize matrices to combine multiple equations into a single equation format. This allows for efficient manipulation and solution finding. In this exercise, we have matrices representing linear systems, which we solve by equating corresponding elements from both sides of the equation.

To begin with, we need to be comfortable with matrix operations, such as addition and subtraction. Whenever you perform these operations, it's crucial to do them element-wise. This means we only add or subtract elements found in the same position across the matrices.

• Each element in the resulting matrix is the sum or difference of elements located at the same row and column position in the original matrices.
• Knowing this helps us simplify the exercise's initial matrix equation.

After simplifying using these rules, the matrices transform into a more straightforward system that can be solved using standard algebraic techniques.

Solving Linear Systems

Solving linear systems involves finding the values of variables that satisfy all equations simultaneously. These systems can be represented as matrices to simplify calculations and demonstrate structure.
In our given problem, equating elements from the transformed matrices generates a set of linear equations like:

• \( x - y = 4 \)
• \( y - x = -4 \)
• \( -y - x = -6 \)
• \( x + y = 6 \)

Each line corresponds to one item in the original system.

By appropriately tackling these equations, we can solve for the variables. Techniques like substitution, elimination, or matrix methods may be utilized. It's beneficial to use substitution when one variable is already isolated, simplifying the system's solving process. For example, rearranging one equation to express \( x \) in terms of \( y \) and then substituting this expression into another equation simplifies the work. Link this to further algebraic manipulation for efficiency.

Algebraic Manipulation

Algebraic manipulation is essential to simplify, rearrange, and solve equations. It involves adding, subtracting, multiplying, or dividing terms to isolate a variable, making it feasible to calculate its value.

This concept played a critical role while solving our exercise. Once the matrix was simplified and linear equations were established, the next step was manipulating these equations to find \(x\) and \(y\).

• Step 1: Isolate a variable. For example, from \( x - y = 4 \), we rearranged to \( x = y + 4 \).
• Step 2: Substitute the expression into another equation. Replace \( x \) in \( x + y = 6 \) with \( y + 4 \).
• Step 3: Simplify the new equation to find one variable, here \( y = 1 \).
• Step 4: Substitute back to find the other variable \(x = 1 + 4 = 5\).

Ensuring that the solutions satisfy check equations solidifies understanding and confirms the solution set's accuracy.