Problem 32 - Algebra 2

Question

Write the matrix equation \(\left[\begin{array}{rr}{2} & {-3} \\ {1} & {4}\end{array}\right] \cdot\left[\begin{array}{l}{r} \\\ {s}\end{array}\right]=\left[\begin{array}{l}{4} \\ {-2}\end{array}\right]\) as a system of linear equations.

Step-by-Step Solution

Answer

The system of linear equations is: \( \{ \begin{array}{l} 2r - 3s = 4 \\ r + 4s = -2 \end{array} \).\)

Step 1: Understand the Matrix Equation

The given matrix equation is \( \begin{bmatrix} 2 & -3 \ 1 & 4 \end{bmatrix} \cdot \begin{bmatrix} r \ s \end{bmatrix} = \begin{bmatrix} 4 \ -2 \end{bmatrix} \). This represents a system of linear equations in which we multiply the matrix on the left by the column vector on the right to get another column vector.

Step 2: Perform the Matrix Multiplication

Multiply the first row of the matrix \( [2, -3] \) with the column vector \( [r, s] \) to form the first equation: \( 2r - 3s = 4 \). Next, multiply the second row of the matrix \( [1, 4] \) with the column vector \( [r, s] \) to form the second equation: \( r + 4s = -2 \).

Step 3: Write the System of Equations

From the results of the matrix multiplication, we get the system of linear equations: \[ \begin{cases} 2r - 3s = 4 \ r + 4s = -2 \end{cases} \]. This represents the same relationships as the original matrix equation, but in a more standard equation format.

Key Concepts

Linear Equations

Linear equations form the building blocks of many mathematical concepts. These are equations of the first degree, which means each variable is raised to the power of one, and they often appear like simple math sentences. For instance, an equation like \( 2r - 3s = 4 \) is considered linear because there is no variable multiplied by another variable, no variable squared, or any complex operations.

• The structure of a linear equation is generally of the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
• Linear equations can describe a straight line when plotted on a graph.
• They are fundamental because they simplify the relationship between variables, making it easier to predict the change in one variable when another variable changes.

With their simplicity, linear equations are the first step towards understanding more intricate mathematical configurations like systems of equations.

Matrix Multiplication

Matrix multiplication is a method that allows us to derive information about how two matrices interact. Unlike multiplying numbers, matrix multiplication is not straightforward due to its rules. It involves taking rows from the first matrix and columns from the second matrix and multiplying them together in a specific way.

In our given exercise scenario:

• The matrix \( \begin{bmatrix} 2 & -3 \ 1 & 4 \end{bmatrix} \) needs to be multiplied by the vector \( \begin{bmatrix} r \ s \end{bmatrix} \).
• This involves multiplying and summing the products of each element of the row of the first matrix with the column of the second vector.
• So for the first row: \( 2 \cdot r + (-3) \cdot s = 4 \).
• For the second row: \( 1 \cdot r + 4 \cdot s = -2 \).

Remember, matrix multiplication is row-wise and column-wise, meaning the orientation and order of matrices matter greatly. Use these principles whenever dealing with matrices in linear algebra.

Systems of Equations

Systems of equations are a set of equations that have multiple variables. The primary goal is to find the values of these variables that satisfy every equation in the system simultaneously. By writing the matrix equation as a system of linear equations, the relationships among variables become much clearer.

• Each equation in a system can derive from scenarios where different relationships among variables exist.
• The system \( \begin{cases} 2r - 3s = 4 \ r + 4s = -2 \end{cases} \) allows us to find the specific values of \( r \) and \( s \) that fulfill both statements.
• There are various methods to solve systems of equations, such as substitution, elimination, or using matrices themselves.

Understanding and solving systems of equations is essential in fields ranging from physics to economics, as it forms the fundamental basis for modeling and solving real-world problems.