Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc}z & 4 r & 8 s \\ 6 p & 2 & 5\end{array}\right]+\left[\begin{array}{ccc}-9 & 8 r & 3 \\ 2 & 5 & 4\end{array}\right]=\left[\begin{array}{ccc}2 & 36 & 27 \\ 20 & 7 & 12 a\end{array}\right]$$

Step-by-Step Solution

Answer

Variables are: \( z = 11 \), \( r = 3 \), \( s = 3 \), and \( p = 3 \). No solution for \( a \) without more information.

Step 1: Understand Matrix Addition

The given problem involves three matrices: two on the left side of the equation that need to be added together, and one on the right side. Matrix addition is performed element-wise, meaning each element in the resulting matrix is the sum of the corresponding elements of the matrices being added.

Step 2: Set Up Element-wise Equations

Write equations based on each position in the matrices:1. For the top-left element: \( z + (-9) = 2 \)2. For the top-center element: \( 4r + 8r = 36 \)3. For the top-right element: \( 8s + 3 = 27 \)4. For the bottom-left element: \( 6p + 2 = 20 \)5. For the bottom-center element: \( 2 + 5 = 7 \)6. For the bottom-right element: \( 5 + 4 = 12a \)

Step 3: Solve for Each Variable

Proceed to solve each of the equations from Step 2 where the variable is present:1. For \( z \): \[z - 9 = 2 \] Add 9 to both sides: \[z = 11 \]2. For \( r \): \[4r + 8r = 36 \] Combine like terms: \[12r = 36 \] Divide by 12: \[r = 3 \]3. For \( s \): \[8s + 3 = 27 \] Subtract 3 from both sides: \[8s = 24 \] Divide by 8: \[s = 3 \]4. For \( p \): \[6p + 2 = 20 \] Subtract 2 from both sides: \[6p = 18 \] Divide by 6: \[p = 3 \]5. Equation for the bottom-center is already satisfied as: \[7 = 7 \]6. For \( a \): \[5 + 4 = 12a \] This simplifies directly as there is no variable present on the left-hand side. Thus, no solution for this position equation as it's incorrect. Possible typo in the original matrix condition.

Key Concepts

Matrix Equations

Matrix equations involve setting up problems where matrices are added, subtracted, or multiplied to find unknown variables. The given exercise focuses on matrix addition to determine the values of unknowns like \( z \), \( r \), \( s \), and \( p \). In matrix equations, each element of the matrices corresponds to a specific position or index.

Matrix Addition Basics:

• Perform element-wise addition, meaning you add each element of one matrix to the corresponding element in the other matrix.
• The result is a new matrix of the same dimension.

When setting up matrix equations:

• Match elements by their position to form individual equations.
• Solve each equation separately to find the values of the unknown variables.

This method is particularly useful in linear algebra to solve systems of equations through matrices.

Systems of Equations

A system of equations consists of multiple equations that have to be solved together to find a common set of variable values. In the context of matrices, a system of equations emerges when more than one variable is present across multiple equations derived from matrix operations.

Solving Matrix Systems:

• Each equation corresponds to a specific position in the matrices involved.
• Solving the system involves addressing each equation individually and finding a solution that satisfies all equations simultaneously.

By substituting the results back into the original matrix equations, you can verify the solutions. This exercise highlights the importance of precision and attention to detail when working with systems derived from matrix arithmetic.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, and matrices among other concepts. It explores linear combinations and transformations, which make it a fundamental element in understanding systems of linear equations.

Connecting Matrices and Linear Algebra:

• Matrices can represent linear transformations, and operations on matrices reflect changes in the system they represent.
• Solving matrix equations, like in the exercise, shows practical applications of linear algebra techniques.

Matrix addition demonstrates a basic, yet crucial operation in linear algebra, emphasizing the integration of linear systems to derive solutions for complex problems.