Problem 39 - Algebra and Trigonometry

Question

Solve for \(x\) and \(y\). $$ \left[\begin{array}{cc}{x} & {2 y} \\ {4} & {6}\end{array}\right]=\left[\begin{array}{cc}{2} & {-2} \\ {2 x} & {-6 y}\end{array}\right] $$

Step-by-Step Solution

Answer

The solution is \( x = 2 \) and \( y = -1 \).

Step 1: Equate corresponding elements

Since the matrices are equal, each corresponding element from one matrix must be equal to the corresponding element in the other matrix. This gives us the set of equations: 1. \( x = 2 \)2. \( 2y = -2 \)3. \( 4 = 2x \)4. \( 6 = -6y \).

Step 2: Solve the first equation

From equation 1, \( x = 2 \). This is already simplified, so we have the solution for \( x \) directly as \( x = 2 \).

Step 3: Solve the second equation

From equation 2, \( 2y = -2 \). Divide both sides by 2 to find \( y = -1 \).

Step 4: Verify the third equation

Checking equation 3 with the found value of \( x = 2 \), we substitute \( x \) to have \( 4 = 2(2) \), which balances as 4 = 4.

Step 5: Verify the fourth equation

Checking equation 4 with the found value of \( y = -1 \), we substitute \( y \) to have \( 6 = -6(-1) \), which balances as 6 = 6.

Key Concepts

Matrix Equality

Matrix equality is a simple yet essential concept in linear algebra. Two matrices are considered equal if and only if they have the same dimensions and each pair of corresponding elements is identical. This means every element in one matrix must match exactly with the element in the same position in the other matrix.
In our exercise, we are given two matrices:

• First Matrix: \( \begin{bmatrix} x & 2y \ 4 & 6 \end{bmatrix} \)
• Second Matrix: \( \begin{bmatrix} 2 & -2 \ 2x & -6y \end{bmatrix} \)

Since these matrices are equal, we can set all their corresponding elements equal to each other. This allows us to form equations by comparing elements from both matrices. Understanding this principle is crucial, as it lays the groundwork for solving systems of equations using matrices. It highlights how matrices can represent and solve equations efficiently, opening up various problem-solving strategies in mathematics. Remember, equality in matrices isn't just visual; it's a precise alignment of all values.

Solving Systems of Equations

Once you have understood matrix equality, you can use this concept to solve systems of equations. In the problem, matching corresponding elements created four separate equations:

• \( x = 2 \)
• \( 2y = -2 \)
• \( 4 = 2x \)
• \( 6 = -6y \)

Each equation is a product of matching the corresponding elements from the two equal matrices. These equations help us find the values of unknowns like \(x\) and \(y\). Start by solving basic equations first (typically the simplest ones) for one variable at a time.
For example:- Solve \( x = 2 \) directly to get \( x = 2 \).- For \( 2y = -2 \), divide both sides by 2 to solve for \( y \), giving \( y = -1 \).Once individual equations are solved, it's important to verify them with other equations to ensure maximum accuracy and consistency.

Equal Matrices

Equal matrices concept combines the principles of matrix equality with the practical application of solving systems of equations. When two matrices are equal, each element pairs in rows and columns align perfectly based on established equations.
In practice, this guarantees that both matrices theoretically represent the same mathematical information. Therefore, they can be used interchangeably in equations once identified as equal, provided all corresponding individual element equations uphold.
In our exercise, by aligning the corresponding elements and solving the resulting equations, we validated that:

• \( x = 2 \) ensures row elements match for both matrices.
• Likewise, \( y = -1 \) fulfills the expectation of equal matrices.

This illustrates an efficient way of approaching problem-solving when faced with complex equations, reiterating that two equal matrices mean more than just visual similarity. They encode identical algebraic information, which can be deciphered by solving their corresponding element equations.