Solving word problems involving matrices, especially those with four variables, can seem daunting. But with a systematic approach and a solid understanding of matrix algebra, you can conquer even the most complex problems. This worksheet focuses on developing your skills in translating real-world scenarios into matrix equations and efficiently solving them. We'll break down the process step-by-step, providing examples and exercises to build your confidence.
Understanding the Fundamentals: Matrices and Systems of Equations
Before diving into word problems, let's refresh our understanding of matrices and their connection to systems of linear equations. A system of four linear equations with four variables (let's say w, x, y, and z) can be represented in matrix form as:
[ a11 a12 a13 a14 ] [ w ] [ b1 ]
[ a21 a22 a23 a24 ] [ x ] = [ b2 ]
[ a31 a32 a33 a34 ] [ y ] [ b3 ]
[ a41 a42 a43 a44 ] [ z ] [ b4 ]
Where:
- A (the coefficient matrix) is a 4x4 matrix containing the coefficients of the variables.
- X (the variable matrix) is a 4x1 matrix containing the variables.
- B (the constant matrix) is a 4x1 matrix containing the constants.
Solving for X involves finding the inverse of matrix A (denoted as A⁻¹) and multiplying it by B:
X = A⁻¹B
This process can be efficiently handled using calculators or software capable of matrix operations. The key is setting up the matrices correctly.
Tackling 4-Variable Word Problems: A Step-by-Step Guide
Let's apply this to a word problem:
Example: A bakery sells four types of pastries: croissants, muffins, scones, and cookies. On Monday, they sold 20 croissants, 15 muffins, 25 scones, and 30 cookies for a total revenue of $250. On Tuesday, they sold 25 croissants, 20 muffins, 15 scones, and 20 cookies for $225. On Wednesday, they sold 15 croissants, 25 muffins, 20 scones, and 15 cookies for $200. On Thursday, they sold 30 croissants, 10 muffins, 10 scones, and 35 cookies for $275. Find the price of each pastry.
Step 1: Define Variables
Let:
- c = price of a croissant
- m = price of a muffin
- s = price of a scone
- k = price of a cookie
Step 2: Set up the Equations
Translate each day's sales into an equation:
- Monday: 20c + 15m + 25s + 30k = 250
- Tuesday: 25c + 20m + 15s + 20k = 225
- Wednesday: 15c + 25m + 20s + 15k = 200
- Thursday: 30c + 10m + 10s + 35k = 275
Step 3: Create the Matrices
Form the coefficient matrix A, variable matrix X, and constant matrix B:
A = [ 20 15 25 30 ]
[ 25 20 15 20 ]
[ 15 25 20 15 ]
[ 30 10 10 35 ]
X = [ c ]
[ m ]
[ s ]
[ k ]
B = [ 250 ]
[ 225 ]
[ 200 ]
[ 275 ]
Step 4: Solve for X
Use a calculator or software to find the inverse of A (A⁻¹) and then calculate X = A⁻¹B. This will give you the values for c, m, s, and k.
Practice Problems
Now it's your turn! Try solving these problems using the step-by-step method outlined above. Remember to clearly define your variables and set up your matrices accurately.
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Investment Portfolio: An investor divides their money among four different accounts: stocks, bonds, mutual funds, and real estate. The returns for each account over three years are known. Set up and solve a matrix equation to find the amount invested in each account. (Create your own realistic data for this problem.)
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Production Planning: A factory produces four types of products. The production time for each product on three different machines is known, as well as the total time available on each machine. Set up and solve a matrix equation to find the number of units of each product that can be produced. (Create your own realistic data for this problem.)
By working through these examples and practice problems, you’ll significantly improve your ability to tackle complex precalculus matrix word problems. Remember, the key lies in a methodical approach, accurate matrix setup, and leveraging technology for efficient calculation.
