Problem 11 - A Graphical Approach to College Algebra

Question

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{rrc} 0 & 5 & x \\ -1 & 3 & y+2 \\ 4 & 1 & z \end{array}\right]=\left[\begin{array}{rrr} 0 & w+3 & 6 \\ -1 & 3 & 0 \\ 4 & 1 & 8 \end{array}\right]$$

Step-by-Step Solution

Answer

x=6, y=-2, z=8, w=2.

Step 1: Compare Matrices Element-wise

To find the values of the variables, compare the corresponding elements of the two matrices. Matrices are equal if and only if all their corresponding elements are equal.

Step 2: Solve for x

Compare the element in the first row, third column of the matrices: \( x = 6 \). Thus, \( x = 6 \).

Step 3: Solve for y

Compare the element in the second row, third column of the matrices: \( y+2 = 0 \). Solving this gives \( y = -2 \).

Step 4: Solve for z

Compare the element in the third row, third column of the matrices: \( z = 8 \). Thus, \( z = 8 \).

Step 5: Solve for w

Compare the element in the first row, second column of the matrices: \( 5 = w+3 \). Solving this gives \( w = 2 \).

Key Concepts

Matrix Element Comparison

When working with matrix equations, comparing elements is a crucial step in solving for variables. Matrix equality is established when each corresponding element between two matrices is identical. In our exercise, we are given two matrices presumed equal, so each element at a respective position in the first matrix equals the element in the same position in the second matrix. Here's how to approach the comparison:

• Align each element according to their rows and columns.
• For every element in the matrix, write down the equation formed by the corresponding elements of the matrices.

By setting the corresponding elements equal to each other, you'll create simple one-variable equations that can be solved independently. This direct comparison simplifies breaking down complex systems into manageable parts.

Systems of Equations

In this context, a matrix equation is equivalent to a system of equations. Each pair of corresponding elements (from the matrices) forms its own equation. Here’s a breakdown of the system from the given matrices:

• The elements in the position (1,3) give us the equation: \( x = 6 \)
• The elements in the position (2,3) form the equation: \( y + 2 = 0 \)
• The elements in the position (3,3) lead to: \( z = 8 \)
• Finally, elements in position (1,2) result in: \( 5 = w + 3 \)

Each equation stands alone and needs to be solved separately. Solving each equation directly gives us the value for each variable, effectively decoding the systems hidden within the matrix form. Systems of equations derived from matrix comparisons are often straightforward due to their isolative nature.

Variable Solving Steps

Solving for variables within matrix elements follows similar rules as solving any simple algebraic equation. Here is a concise step-by-step guide:1. **Identify Equations**: Each element equals another element, forming simple equations (e.g., from the matrix comparison, \( x = 6 \)). 2. **Isolate Variables**: - Begin with the simplest form of equation. For instance, if \( x = 6 \), you have already found "x". - For equations like \( y + 2 = 0 \), rearrange them to isolate "y" by subtracting 2 from both sides, yielding \( y = -2 \).3. **Solve Sequentially**: Move methodically through each variable. Ensure to cross-check each variable by substituting back into its original equation to verify correctness. By carefully following these steps, you'll arrive at the solutions effectively and accurately. The process not only boosts problem-solving skills but also deepens understanding of foundational algebraic principles.