Problem 43 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Equate the first elements of the matrices

Start by equating the element in the first row and first column of both matrices: \[ x = 2. \] This gives us the value for \(x\) immediately.

Step 2: Equate the second elements of the first row

Now, equate the second elements in the first row of both matrices: \[ 2y = -2. \] Solve for \(y\) by dividing both sides by 2: \[ y = -1. \]

Step 3: Verify with the lower half of the matrices

From Step 1, we know \(x = 2\). Check the second matrix's lower half using this value: \[ 4 = 2(2) \] which is \(4\). Next, using \(y = -1\): \[ 6 = -6(-1). \] Both are true; thus, \(x = 2\) and \(y = -1\) satisfy the entire matrix equality.

Key Concepts

Solving Matrices

The concept of solving matrices involves finding the unknown variables within a given set of matrices. Solving matrix equations is very similar to solving equations in algebra, but instead, you work within a structured grid format.
We approach this by equaling the corresponding elements in each matrix. In our original exercise, we start with two matrices, and our task is to find the values of the unknown variables like \(x\) and \(y\).

• Firstly, identify which parts of the matrices are directly comparable. Each element in the matrix has a specific location, like a puzzle piece, and must match the other matrix exactly.
• By focusing on one element at a time, such as starting with the top left, you directly compare each part of the two matrices.
• Once the elements match, you solve for the unknowns using basic algebraic techniques.

This process of solving matrices ensures that the equation holds true overall, aligning each piece to its counterpart.

Matrix Equality

Matrix equality is a principle that states two matrices are equal if and only if their corresponding entries are equal.
When solving for unknowns in matrix equations, it's important to understand this basis:

• Each corresponding element in both matrices must be identically equal for the matrices themselves to be equal.
• In our example, each entry, such as \( x = 2 \), directly supports the principle of matrix equality.

Matrix equality is fundamental because it ensures all dimensions and values match perfectly, which leads to accurate solutions. By applying matrix equality, each part of the solution must confirm the equivalency of corresponding elements across all rows and columns.

System of Equations

At the heart of matrix problems solved through matrix equality lies a system of equations.
Each matrix element comparison forms an equation that must be satisfied.

• In our specific example, equating the elements in the first row forms two equations: \( x = 2 \) and \( 2y = -2 \).
• Adjust and solve these equations using standard algebraic methods to find the values for the variables.

A system of equations constructed from a matrix will often require the equations to be compatible, ensuring a consistent solution across all equations.
Through solving, you verify the values, such as \( x = 2 \) and \( y = -1 \), satisfy the entire set of conditions established by the matrix equality, ensuring a coherent and consistent resolution for all involved variables.