Problem 39 - Precalculus: Mathematical for Calculus

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

\(x = 2\), \(y = -1\).

Step 1: Set Up the Matrix Equation

The problem gives us two matrices that are equal. This means we can set the corresponding elements equal to each other. Let's write down the equations for each pair of elements:1. The first element in the first row: \(x = 2\)2. The second element in the first row: \(2y = -2\)3. The first element in the second row: \(4 = 2x\)4. The second element in the second row: \(6 = -6y\)

Step 2: Solve for x

Starting with the equations involving \(x\):\(x = 2\) and \(4 = 2x\).From the first equation, we have \(x = 2\), which already satisfies the second equation \(4 = 2 \times 2\). Thus, \(x = 2\) is consistent.

Step 3: Solve for y

Now solve the equations involving \(y\):From \(2y = -2\), divide both sides by 2 to get: \(y = -1\).Verify with the second equation involving \(y\):6 = -6y Substitute \(y = -1\):\(6 = -6(-1) = 6\), which is correct. Thus, \(y = -1\) is consistent.

Key Concepts

Matrix Equality

Matrix equality is a fundamental concept in mathematics that tells us when two matrices are exactly the same. This is pivotal when working with matrix equations. For two matrices to be equal, every corresponding element must be the same.
In this exercise, we had the matrix equation:\[\begin{bmatrix}x & 2y \4 & 6\end{bmatrix} = \begin{bmatrix}2 & -2 \2x & -6y\end{bmatrix}\]By directly comparing each position in these matrices:

• The element at the first row, first column tells us that \(x = 2\).
• The element at the first row, second column gives us \(2y = -2\).
• In the second row, for the first column, \(4 = 2x\) follows.
• The element in the second row, second column provides \(6 = -6y\).

Successfully setting these elements equal is the essence of matrix equality. This allows us to create a system of equations that can be independently solved.

Matrix Algebra

Matrix algebra involves various operations with matrices, much like normal algebra. It includes addition, subtraction, multiplication, and often solving equations using techniques specific to matrices.
When dealing with matrices, each operation has its rules. Here, our primary goal was to understand how equations derived from matrix equality can be solved:

• We compared matrices to form equations: like \(x = 2\) and \(2y = -2\).
• We then used basic algebraic operations to solve for variables.

In more complex scenarios, you might also encounter matrix multiplication and the need to find determinants or inverses. For this particular problem, focusing on matrix component equality led directly to solving equations, which is an essential part of matrix algebra.

Solving Linear Equations

Solving linear equations involves finding the values of unknown variables that satisfy given equations. In this matrix exercise:

• We established that \(x = 2\) directly connects to the elements of the matrices.
• For the equation \(2y = -2\), we isolated \(y\) by dividing both sides by 2, resulting in \(y = -1\).

Verification is key in solving equations. After finding values for \(x\) and \(y\), confirming them in the original matrix equations ensures consistency and correctness.
Sometimes, matrix-based equations might require additional tools like substitution or elimination to solve. Ensuring each step adheres to basic algebraic rules is crucial.Leveraging matrix equality simplifies the complexity, providing a straightforward path to the solution.