Problem 9 - A Graphical Approach to College Algebra

Question

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ll} w & x \\ y & z \end{array}\right]=\left[\begin{array}{rr} 3 & 2 \\ -1 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

\(w = 3, x = 2, y = -1, z = 4\)

Step 1: Understand the Equality of Matrices

To find the value of each variable in the matrix, we need to equate the corresponding elements from both matrices. The matrices are equal if and only if their corresponding elements are equal. In this case, this gives us four equations: \( w = 3 \), \( x = 2 \), \( y = -1 \), and \( z = 4 \).

Step 2: Solve for Each Variable

From the equations derived from the equality of matrices, solve for each variable. \( w = 3 \) implies that the value of \( w \) is 3. Similarly, \( x = 2 \) means \( x \) is 2. For \( y = -1 \), the value of \( y \) is -1. Finally, \( z = 4 \) tells us that \( z \) is 4.

Step 3: Confirm the Solution

Check the solution by substituting the values back into the original matrix and comparing it with the given matrix. If they are equal, then our values are correct. Substituting gives us \( \begin{bmatrix} 3 & 2 \ -1 & 4 \end{bmatrix} \), which matches the given matrix.

Key Concepts

Equality of Matrices

When dealing with matrices, understanding the principle of equality is essential. Matrices are equal if and only if they have the same dimensions and their corresponding elements are exactly the same. For example, consider two matrices: \( A \) and \( B \). For these matrices to be equal, not only must they each have the same number of rows and columns, but each element \( a_{ij} \) in matrix \( A \) must equal the corresponding element \( b_{ij} \) in matrix \( B \).
This means that if we compare two matrices and find even one pair of corresponding elements to be different, the matrices are not equal.

• Check dimensions first: make sure both matrices are of the same size.
• Compare each corresponding element: every element must match exactly.

Understanding this foundational concept is the first step in solving matrix equations.

Matrix Variables

Matrix variables often represent unknown values within a matrix that can be solved or deciphered by equating the matrices. In our exercise, the matrix variables are simply the symbols \( w, x, y, \) and \( z \). These represent unknown numbers that we need to find.
Solving for matrix variables involves setting each variable equal to its corresponding number in the other matrix. In other words, you establish equations based on the equality of elements.

• Assign: Based on the matrix \( \begin{bmatrix} w & x \ y & z \end{bmatrix} \), you equate each entry to the respective number in \( \begin{bmatrix} 3 & 2 \ -1 & 4 \end{bmatrix} \).
• Equations: This leads us to: \( w = 3 \), \( x = 2 \), \( y = -1 \), and \( z = 4 \).
• Solve: Simply solve these basic equations to determine the values of the variables.

This approach helps in determining each variable's value independently and ensures that all unknowns are addressed correctly.

Matrix Solution Verification

After determining the values of the matrix variables, verifying these solutions is a crucial step. Solution verification ensures that you haven't made any errors and your calculations are correct. Here's how you do it for matrix equations:First, substitute the found values back into the original matrix of variables. So, replace \( w, x, y, \) and \( z \) with 3, 2, -1, and 4 respectively. This gives you a matrix like this:\[\begin{bmatrix}3 & 2 \-1 & 4\end{bmatrix}\]Next, compare this constructed matrix to the given matrix \( \begin{bmatrix} 3 & 2 \ -1 & 4 \end{bmatrix} \). If every element matches exactly, then your solution is verified as correct.

• Substitute: Replace each variable with the corresponding value you solved for.
• Compare: Ensure every element in the new matrix aligns with the original matrix provided.

Through this simple check, ensure that no step was missed and the solution maintains its integrity.