Problem 45 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The provided matrix equation is \( 2\begin{bmatrix}x & y \ x+y & x-y \end{bmatrix} = \begin{bmatrix}2 & -4 \ -2 & 6 \end{bmatrix} \). We need to solve for \(x\) and \(y\) such that both matrices are equal.

Step 2: Equate Elements of the Matrices

For two matrices to be equal, their corresponding elements must be equal. So, from the equation \( 2\begin{bmatrix}x & y \ x+y & x-y \end{bmatrix} = \begin{bmatrix}2 & -4 \ -2 & 6 \end{bmatrix} \), we get:1. \(2x = 2\)2. \(2y = -4\)3. \(2(x+y) = -2\)4. \(2(x-y) = 6\)

Step 3: Solve the First Pair

Solve the equations \(2x = 2\) and \(2y = -4\) to find \(x\) and \(y\).- Dividing the first equation by 2: \[x = \frac{2}{2} = 1\]- Dividing the second equation by 2: \[y = \frac{-4}{2} = -2\]

Step 4: Verify with the Remaining Conditions

Substitute \(x = 1\) and \(y = -2\) into the remaining equations to verify the solution:- For \(2(x+y) = -2\): \[2(1 + (-2)) = 2(-1) = -2 \text{ (True)}\]- For \(2(x-y) = 6\): \[2(1 - (-2)) = 2(3) = 6 \text{ (True)}\]

Step 5: Conclude the Solution

Since both conditions are met and all calculations check out, the solution to the matrix equation is \( x = 1 \) and \( y = -2 \).

Key Concepts

Solving Systems of Equations

When tackling systems of equations, we seek values for variables that simultaneously satisfy multiple equations. In the realm of matrices, this concept extends naturally. Here, we engage with a system represented not just by equations but by matrices. The goal is to find the right values of our variables that make each matrix element pair equal. For instance, given a matrix equation:\[2\begin{bmatrix}x & y\x+y & x-y\end{bmatrix} = \begin{bmatrix}2 & -4\-2 & 6\end{bmatrix}\]To find solutions for \(x\) and \(y\), break the task into smaller parts based on matrix cells:

• Solve \(2x = 2\) to find \(x\)
• Solve \(2y = -4\) to find \(y\)
• Check with \(2(x+y) = -2\) and \(2(x-y) = 6\)

By addressing these equations individually, you'll not only solve for \(x\) and \(y\) but also ensure that the solutions harmonize with all provided conditions, acquiring a deeper understanding of the relevance between equations and matrices.

Matrix Equality

Matrix equality is a straightforward yet powerful concept. Two matrices are considered equal when all of their corresponding elements are identical. Consequently, this principle allows us to derive relationships between unknowns within a mathematical or real-world context from matrix representations.Consider the current problem's matrices:\[2\begin{bmatrix}x & y\x+y & x-y\end{bmatrix} = \begin{bmatrix}2 & -4\-2 & 6\end{bmatrix}\]We equate corresponding elements:

• First row, first column: \(2x = 2\)
• First row, second column: \(2y = -4\)
• Second row, first column: \(2(x+y) = -2\)
• Second row, second column: \(2(x-y) = 6\)

By setting each element of the left matrix equal to the corresponding element of the right matrix, we get manageable equations to solve. This method highlights the utility of matrix equations in simplifying and organizing complex systems of equations, transforming them into decodable chunks.

Precalculus

Precalculus typically serves as the foundation for calculus, involving a range of mathematical concepts, including functions, series, and, crucially, matrices. In this context, matrices aren't simply static collections of numbers; they're dynamic tools that model systems encountered in calculus and beyond. Matrix equations interconnect different areas of precalculus—like algebra and geometry—by offering a structured way to solve multiple equations simultaneously. In preparation for calculus, students learn to:

• Manipulate matrices similar to arithmetic operations
• Solve matrix equations as in algebraic contexts
• Apply matrix equality properties across varied problems

These skills enable a smooth transition to calculus and higher-level mathematics, emphasizing the precision and elegance of mathematical expressions. Matrix-related problems nurture critical thinking, allowing students to grasp more abstract concepts by breaking them into methodical, logical steps.