Problem 11 - Algebra 2

Question

Write a matrix equation for each system of equations. \(3 m-7 n=-43\) \(6 m+5 n=-10\)

Step-by-Step Solution

Answer

Matrix equation: \( \begin{bmatrix} 3 & -7 \\ 6 & 5 \end{bmatrix} \begin{bmatrix} m \\ n \end{bmatrix} = \begin{bmatrix} -43 \\ -10 \end{bmatrix} \).

Step 1: Write the System of Equations

We have two equations given in the problem:1. \( 3m - 7n = -43 \)2. \( 6m + 5n = -10 \).

Step 2: Identify Variables and Coefficients

In the system of equations, the variables are \( m \) and \( n \). The coefficients for the first equation are 3 and -7, and for the second equation are 6 and 5.

Step 3: Arrange Coefficients into a Matrix

Take the coefficients of \( m \) and \( n \) from both equations and form them into a matrix:\[\begin{bmatrix} 3 & -7 \ 6 & 5 \end{bmatrix}\].

Step 4: Arrange Variables into a Column Matrix

The variables \( m \) and \( n \) are written as a column matrix:\[\begin{bmatrix} m \ n \end{bmatrix}\].

Step 5: Arrange Constants into a Column Matrix

The constants from the equations, \(-43\) and \(-10\), form another column matrix:\[\begin{bmatrix} -43 \ -10 \end{bmatrix}\].

Step 6: Write the Matrix Equation

Combine the matrices from Steps 3, 4, and 5 to form the matrix equation:\[\begin{bmatrix} 3 & -7 \ 6 & 5 \end{bmatrix} \begin{bmatrix} m \ n \end{bmatrix} = \begin{bmatrix} -43 \ -10 \end{bmatrix}\].

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with a common set of variables. In our case, the system consists of the equations \(3m - 7n = -43\) and \(6m + 5n = -10\). Each equation contains two variables, \(m\) and \(n\). The ultimate goal is to find the values for these variables that satisfy all the equations simultaneously.

• Equations are usually presented in a manner where variables are on one side and constants are on the other side.
• The solution to a system is the point where each line described by these equations intersect.
• Multiple methods exist to solve systems, such as substitution, elimination, and using matrices.

Understanding systems of equations is foundational in algebra and essential for more advanced topics in mathematics and applied science.

Matrix Representation

Matrix representation involves arranging the coefficients and variables of a system of equations into a structured format known as a matrix. This format makes the system of equations easier to manipulate and solve using various algebraic techniques.

• Coefficients from the equations are placed into a matrix, each row representing an equation, and each column representing a variable.
• In our step-by-step solution, a matrix is formed from the coefficients: \[\begin{bmatrix} 3 & -7 \ 6 & 5 \end{bmatrix}\]
• Matrices simplify complex operations and help visual comparisons among coefficients and constants.

By translating equations into a matrix, we not only standardize the approach for solving systems but also set the stage for using sophisticated tools and methodologies.

Algebraic Equations

Algebraic equations form the core of many mathematical solutions and are expressions equated to a number. In a system, these algebraic forms condense into equations showing relationships between variables. For instance, our given equations are algebraic in form and can be solved to find the values of \(m\) and \(n\).

• Equations can include operations like addition, subtraction, multiplication, and division.
• The expression can have constants (numbers) and variables (letters representing numbers).
• Simplifying and manipulating equations using algebraic rules is essential to isolate variables and solve for them.

Grasping how to form and solve algebraic equations is critical in learning how to approach more challenging problems in math and science.

Linear Algebra

Linear algebra deals extensively with vectors, matrices, systems of equations, and their transformations. It's a branch of mathematics useful for understanding and solving systems of linear equations, such as the one presented in our exercise.

• Linear algebra provides the framework to transform and solve matrix equations.
• Techniques such as matrix multiplication, inversion, and determinants are powerful tools in analyzing systems.
• This field has applications in computer graphics, engineering, physics, and economics.

Learning linear algebra equips students with a robust set of skills to analyze complex systems and solve real-world mathematical challenges effectively.