Problem 37 - Algebra and Trigonometry

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: Set Up the Matrix Equation

We are given the equation \( 2\begin{bmatrix} x & y \ x+y & x-y \end{bmatrix} = \begin{bmatrix} 2 & -4 \ -2 & 6 \end{bmatrix} \). To solve for \( x \) and \( y \), we first need to simplify and compare the matrices.

Step 2: Simplify the Left Side

Distribute the scalar 2 to each element in the matrix on the left side: \[ 2\begin{bmatrix} x & y \ x+y & x-y \end{bmatrix} = \begin{bmatrix} 2x & 2y \ 2(x+y) & 2(x-y) \end{bmatrix} \].

Step 3: Compare the Matrices

Set the corresponding elements from both matrices equal: 1. \(2x = 2\) 2. \(2y = -4\) 3. \(2(x+y) = -2\) 4. \(2(x-y) = 6\)

Step 4: Solve for \(x\) and \(y\) from Equations 1 and 2

For Equation 1, divide both sides by 2: \( 2x = 2 \Rightarrow x = 1 \). For Equation 2, divide both sides by 2: \( 2y = -4 \Rightarrow y = -2 \).

Step 5: Verify Solutions with Equations 3 and 4

Substitute \( x = 1 \) and \( y = -2 \) into Equations 3 and 4 to check consistency. - For Equation 3: \( 2(1 + (-2)) = -2 \Rightarrow 2(-1) = -2 \) is true.- For Equation 4: \( 2(1 - (-2)) = 6 \Rightarrow 2(3) = 6 \) is true.

Key Concepts

Solving Simultaneous Equations

Solving simultaneous equations involves finding the values of the variables that satisfy all equations at the same time. In the context of our matrix problem, we had simultaneous equations for each corresponding element of the matrices. This means each position in the matrix provides a separate equation which must be satisfied by the same values of the variables.

• First, identify each equation from corresponding elements.
• Solve one by one to find the values of variables such as \(x\) and \(y\).
• Check solution consistency by substituting back into the original equations.

Begin with simpler equations to find values quickly and easily verify with the more complex equations from the matrix.

Matrix Comparison

Matrix comparison consists of setting elements with the same position in two matrices equal to each other. This approach is useful when dealing with matrix equations, as it allows us to derive a system of simultaneous equations that we can solve.

• Ensure both matrices are of the same size for comparison.
• Identify corresponding elements that should be equal.
• Derive equations for each element pair that will guide you to finding the variables' values.

This step-by-step inspection facilitates further simplification and reveals whether a solution is possible and correct.

Scalar Multiplication in Matrices

Scalar multiplication in matrices involves distributing a single number (which is the scalar) across all elements of the matrix. It's one of the basic operations that keep the transformation of equation straightforward and manageable. Conceptually:

• Take each element in the original matrix.
• Multiply it by the scalar to adjust the matrix dimensions.
• Form the resulting matrix to be used in further computations or comparisons.

In essence, this operation might transform the hierarchical relationship within a matrix, but it keeps the relative values constant, simplifying further arithmetics on matrices.