The triangle has a perimeter of 30​ cm, and y is twice the length of x.
a. Write a system of equations to represent this situation.
b. Write a matrix to represent the system of equations you wrote in part a.
c. Find the reduced row echelon form of the matrix you wrote in part b.
d. What does the third column of the matrix you found in part c ​represent?\

Let's break down the problem step by step.

### Given Information:
- The perimeter of the triangle is \( 30 \) cm.
- \( y \) is twice the length of \( x \), so \( y = 2x \).

Let's denote the sides of the triangle as \( x \), \( y \), and \( z \).

### Part a: System of Equations
We need to write a system of equations that represents the situation.

1. The equation for the perimeter of the triangle:
   \[
   x + y + z = 30
   \]
2. The relationship between \( y \) and \( x \):
   \[
   y = 2x
   \]

So, the system of equations is:
\[
\begin{aligned}
1. & \quad x + y + z = 30 \\
2. & \quad y - 2x = 0
\end{aligned}
\]

### Part b: Matrix Representation
Next, we write a matrix to represent this system of equations. Let's express the system in matrix form \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the column vector of variables, and \( B \) is the column vector of constants.

The system of equations in standard form is:

\[
\begin{aligned}
1. & \quad x + y + z = 30 \\
2. & \quad -2x + y + 0z = 0
\end{aligned}
\]

This corresponds to the following matrix:

\[
\begin{pmatrix}
1 & 1 & 1 \\
-2 & 1 & 0
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
=
\begin{pmatrix}
30 \\
0
\end{pmatrix}
\]

### Part c: Reduced Row Echelon Form (RREF)
To find the RREF of the matrix, we'll first construct the augmented matrix and then perform row operations.

The augmented matrix is:

\[
\begin{pmatrix}
1 & 1 & 1 & \vert & 30 \\
-2 & 1 & 0 & \vert & 0
\end{pmatrix}
\]

Let's find the RREF.

\[
\text{RREF:} \begin{pmatrix} 1 & 0 & \frac{1}{3} & \vert & 10 \\ 0 & 1 & \frac{2}{3} & \vert & 20 \end{pmatrix}
\]

### Part d: Interpretation of the Third Column
The third column of the matrix in RREF corresponds to the variable \( z \). The entries in the third column ( \( \frac{1}{3} \) and \( \frac{2}{3} \) ) indicate the relationship of \( z \) with respect to \( x \) and \( y \) in the solution of the system.

This means:
- \( z = \frac{1}{3} \times 10 + \frac{2}{3} \times 20 \), or more generally, \( z = 10x + 20y \).

### Do you want further details or have any questions?

Here are 5 relative questions that build upon the previous information:
1. How do you interpret the relationship between the sides \( x \), \( y \), and \( z \) after solving the system of equations?
2. Can you determine the specific values of \( x \), \( y \), and \( z \) from the RREF matrix?
3. How would the RREF matrix change if the perimeter were different, say 40 cm?
4. How would the solution change if \( y \) were three times \( x \) instead of two?
5. What role does the third column play in identifying potential values of \( z \) given different perimeter conditions?

**Tip:** When working with systems of linear equations, the RREF of a matrix provides the simplest form to interpret the solutions directly, making it easier to identify relationships between variables.

Learn how to solve a triangle perimeter problem using systems of equations. Understand the relationship between triangle sides, represented by variables x, y, and z. Includes step-by-step calculations of matrix operations and Reduced Row Echelon Form (RREF). Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation