Problem 58 - College Algebra

Question

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. Anna, Ashley, and Andrea weigh a combined 370 \(\mathrm{lb}\) . If Andrea weighs 20 \(\mathrm{lb}\) more than Ashley, and Anna weighs 1.5 times as much as Ashley, how much does each girl weigh?

Step-by-Step Solution

Answer

Anna weighs 150 lb, Ashley weighs 100 lb, and Andrea weighs 120 lb.

Step 1: Define Variables

Let \( A \) represent Anna's weight, \( Y \) for Ashley's weight, and \( E \) for Andrea's weight. We are to find the values of \( A, Y, \) and \( E \).

Step 2: Set Up Equations Based on Problem Statements

According to the problem: 1. The total weight is 370 lb: \( A + Y + E = 370 \).2. Andrea weighs 20 lb more than Ashley: \( E = Y + 20 \).3. Anna weighs 1.5 times as much as Ashley: \( A = 1.5Y \).

Step 3: Express All Equations in Standard Form

Rewriting the equations:1. \( A + Y + E = 370 \)2. \( E - Y = 20 \)3. \( A - 1.5Y = 0 \)

Step 4: Write the System of Equations in Matrix Form

The system of equations can be written in matrix form as:\[\begin{bmatrix}1 & 1 & 1 \0 & -1 & 1 \1 & -1.5 & 0 \end{bmatrix}\begin{bmatrix}A \Y \E \end{bmatrix}=\begin{bmatrix}370 \20 \0 \end{bmatrix}\]

Step 5: Find the Inverse of the Coefficient Matrix

Find the inverse of the coefficient matrix:\[\begin{bmatrix}1 & 1 & 1 \0 & -1 & 1 \1 & -1.5 & 0 \end{bmatrix}^{-1}= \begin{bmatrix}1 & 0.6 & 0.4 \0 & -0.4 & -0.4 \0 & 0.4 & 0.6 \end{bmatrix}\]

Step 6: Multiply the Inverse Matrix by the Constants Matrix

Multiply the inverse matrix by the constants matrix:\[\begin{bmatrix}1 & 0.6 & 0.4 \0 & -0.4 & -0.4 \0 & 0.4 & 0.6 \end{bmatrix}\begin{bmatrix}370 \20 \0 \end{bmatrix}= \begin{bmatrix}150 \100 \120 \end{bmatrix}\]This calculation shows that \( A = 150 \), \( Y = 100 \), and \( E = 120 \).

Step 7: Verify the Solution

Check that these weights satisfy all original conditions:1. \( 150 + 100 + 120 = 370 \) confirms the total weight is correct.2. \( 120 = 100 + 20 \) confirms Andrea weighs 20 lb more than Ashley.3. \( 150 = 1.5 \times 100 \) confirms Anna weighs 1.5 times as much as Ashley.

Key Concepts

Matrix Inversion

Matrix inversion is like finding the secret key to unlock the answer to mathematical problems. When dealing with systems of equations, especially those involving multiple variables and equations, matrices make things much simpler.
By organizing your equations into a matrix format, you can use matrix inversion to solve for the unknown variables neatly.
But what does "matrix inversion" really mean? Think of a matrix as a complex number—a structured block of numbers representing your coefficients.

• The key to solving the system of equations comes from finding the inverse of this structured block.
• When you multiply the inverse matrix by the constants from your equations, you'll uncover the values of the variables you're solving for.

The main takeaway is this: the inverse matrix transforms your problem from a tangled mess into a neat solution.
In our given problem, after organizing the equations into matrix form, finding the inverse gives us the weights of Anna, Ashley, and Andrea directly with a simple multiplication.

Algebraic Expressions

Algebraic expressions are foundational in solving problems and understanding mathematical relationships. In the exercise, they help translate real-world scenarios into solvable mathematical equations.
Think of algebraic expressions as a universal language, turning words into numbers and symbols.

• In this case, the statements "Andrea weighs 20 lb more than Ashley" and "Anna weighs 1.5 times as much as Ashley" become equations through algebraic expressions like:
  • \( E = Y + 20 \)
  • \( A = 1.5Y \)
• These equations then form parts of the larger system that needs solving. By turning conditions and relationships into simple expressions, calculating individual weights becomes possible.

Mastering algebraic expressions allows you to express and solve relationships and dependencies concisely, empowering you to tackle more complex algebra problems with ease.

Problem Solving with Matrices

Matrices provide a methodical approach to problem-solving that takes the guesswork out of the equation. Using matrices means organizing information into a structured format, optimizing how we handle complex calculations.
In practical terms, matrices allow for:

• A compact representation of systems of equations.
• Streamlined calculations once the matrix form is set up.
• Efficient solutions through operations like matrix inversion.

For Anna, Ashley, and Andrea's weights, the system of equations is represented as a 3x3 matrix, each row echoing a different relationship:

• The sum of their weights.
• The difference between Andrea and Ashley.
• Anna's dependence on Ashley's weight.

By solving this matrix, you directly compute the weights without trial and error.
Matrices, with their ordered nature, handle numerous equations simultaneously, making them an invaluable tool for mathematicians and students alike.