📚 Introduction to Systems of Equations and Matrices for Word Problems
Many real-world problems can be modeled using systems of equations. Representing these systems with matrices allows us to solve them efficiently, especially when dealing with multiple variables. This guide provides a comprehensive overview of how to tackle word problems using this powerful approach.
📜 A Brief History
The concept of solving simultaneous equations dates back to ancient civilizations. However, the systematic use of matrices to solve these equations gained prominence in the 19th century, thanks to mathematicians like Arthur Cayley.
🔑 Key Principles
- 🔍 Defining Variables: The first crucial step is to identify and clearly define the variables involved in the problem. Assign a letter (e.g., x, y, z) to each unknown quantity.
- 📝 Translating Words into Equations: Carefully translate the information provided in the word problem into mathematical equations. Look for key phrases that indicate relationships between the variables (e.g., 'sum', 'difference', 'twice', 'is equal to').
- 🔢 Forming a System of Equations: Combine the equations to form a system. The number of independent equations should ideally match the number of variables to ensure a unique solution.
- 📊 Representing as a Matrix: Represent the system of equations in matrix form: $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix.
- ➗ Solving the Matrix Equation: Use matrix operations (e.g., Gaussian elimination, matrix inversion) to solve for the variable matrix $X$. If $A$ is invertible, $X = A^{-1}B$.
- ✅ Interpreting the Solution: Once you have the values of the variables, interpret them in the context of the original word problem to provide a meaningful answer.
🌍 Real-World Examples
Example 1: Investment Problem
An investor has \$20,000 to invest in three different funds: a money-market account paying 3\% annual interest, municipal bonds paying 4\% annual interest, and mutual funds paying 7\% annual interest. The investor wants to invest twice as much in municipal bonds as in the money-market account. How much should be invested in each type of fund if the investment goal is to earn \$1,100 in annual interest?
Let $x$ be the amount invested in the money-market account, $y$ be the amount invested in municipal bonds, and $z$ be the amount invested in mutual funds.
We can set up the following system of equations:
- 💰Total investment: $x + y + z = 20000$
- 📈Interest earned: $0.03x + 0.04y + 0.07z = 1100$
- 🤝Relationship between investments: $y = 2x$
Rewrite the third equation: $-2x + y = 0$
The matrix form of the system is:
$\begin{bmatrix} 1 & 1 & 1 \\ 0.03 & 0.04 & 0.07 \\ -2 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 20000 \\ 1100 \\ 0 \end{bmatrix}$
Solving this system using Gaussian elimination or matrix inversion yields:
$x = 4000$, $y = 8000$, $z = 8000$
Therefore, the investor should invest \$4,000 in the money-market account, \$8,000 in municipal bonds, and \$8,000 in mutual funds.
Example 2: Mixture Problem
A chemist needs to mix a 20\% acid solution with a 50\% acid solution to obtain 100 ml of a 32\% acid solution. How many milliliters of each solution should the chemist use?
Let $x$ be the amount of the 20\% solution and $y$ be the amount of the 50\% solution.
We can set up the following system of equations:
- 🧪Total volume: $x + y = 100$
- 🧪Acid concentration: $0.20x + 0.50y = 0.32(100)$ which simplifies to $0.20x + 0.50y = 32$
The matrix form of the system is:
$\begin{bmatrix} 1 & 1 \\ 0.20 & 0.50 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 100 \\ 32 \end{bmatrix}$
Solving this system yields:
$x = 60$, $y = 40$
Therefore, the chemist should use 60 ml of the 20\% solution and 40 ml of the 50\% solution.
💡 Conclusion
Solving word problems using systems of equations and matrices is a powerful technique applicable in various fields. By carefully defining variables, translating information into equations, and utilizing matrix operations, complex problems can be solved systematically.