Solving a System of Linear Equations Using Matrix Methods

Math Problem Statement

Question content area top Part 1 Use technology to find the solution of the following system. 1.6x minus− 0.7y plus+ 2.4z equals= 4.67 3.2x minus− 5.3y plus+ 7.1z equals= 14.97 1.8y plus+ 5.93 equals= 3.2z minus−0.3x Question content area bottom Part 1 The solution set is left parenthesis nothing comma nothing comma nothing right parenthesisenter your response here,enter your response here,enter your response here. (Simplify your answers. Type integers or decimals.)

Solution

We have a system of three equations to solve. The system is as follows:

1.6x - 0.7y + 2.4z &= 4.67 \\ 3.2x - 5.3y + 7.1z &= 14.97 \\ 1.8y + 5.93 &= 3.2z - 0.3x \end{aligned}$$ ### Step 1: Rearrange the third equation We will first simplify the third equation to make it more comparable to the other two: $$1.8y + 5.93 = 3.2z - 0.3x$$ Move all terms to one side of the equation: $$0.3x + 1.8y - 3.2z = -5.93$$ Now, our system of equations becomes: $$\begin{aligned} 1.6x - 0.7y + 2.4z &= 4.67 \\ 3.2x - 5.3y + 7.1z &= 14.97 \\ 0.3x + 1.8y - 3.2z &= -5.93 \end{aligned}$$ ### Step 2: Solve using matrix methods or substitution We can represent this system as a matrix and solve it using technology (e.g., a calculator or software like Python or MATLAB). I'll use Python here to find the solution. Let me calculate that for you.The solution to the system of equations is: $$x = 0.3, \quad y = -0.5, \quad z = 1.6$$ ### Final Solution Set: $$(0.3, -0.5, 1.6)$$ Would you like more details on any of these steps or explanations? Here are 5 related questions for further exploration: 1. How do you represent a system of equations as a matrix? 2. What are some methods for solving systems of linear equations besides substitution? 3. How can matrix inversion help in solving systems of linear equations? 4. How does the Gaussian elimination method work for solving linear systems? 5. How does technology like Python or MATLAB simplify solving large systems of equations? **Tip:** When solving systems of equations with decimals, using matrix methods or computational tools helps ensure accuracy and saves time.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Matrix Methods

Formulas

Ax = B (matrix equation)
Matrix inversion
Gaussian elimination

Theorems

Matrix Inversion Theorem
Gaussian Elimination

Suitable Grade Level

Grades 10-12

Related Recommendation

Solve Systems of Linear Equations Using the Matrix Method

Solve Linear Equations Using Matrix Method: 9y + 2z = 18, 3x + 2y + z = 1, x - y = -3

Solving Systems of Equations Using Matrix Inversion: x - 2y - z = 7, x + 4y + z = 7, 3x + 5y + 2z = 7

Solving a System of Linear Equations using Gaussian Elimination