Solving a 3-Equation Linear System: 2x + 2y + 3z = 3, x + 3y + 2z = 5, 3x + y + z = 5

Math Problem Statement

Solve the system of three linear equations: 1. 2x + 2y + 3z = 3 2. x + 3y + 2z = 5 3. 3x + y + z = 5

The image you uploaded contains a system of three linear equations:

Let's solve this system of equations step by step using substitution or elimination.

The system can be written as a matrix equation $A \mathbf{x} = \mathbf{b}$ , where:

2 & 2 & 3 \\ 1 & 3 & 2 \\ 3 & 1 & 1 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 3 \\ 5 \\ 5 \end{pmatrix}$$ ### Step 2: Solving by substitution/elimination: - From Equation (2): $$x = 5 - 3y - 2z$$ Substitute this into the other two equations (1) and (3) and solve for $$y$$ and $$z$$. After solving this system, the solution is: $$x = 1, \quad y = 1, \quad z = 1$$ So, the solution to the system is $$(x, y, z) = (1, 1, 1)$$. --- Would you like more details on how this was solved step-by-step, or do you have any other questions? Here are 5 related questions to expand on this topic: 1. How can systems of equations be solved using matrix methods like Gaussian elimination? 2. What are the conditions for a system of equations to have a unique solution, no solution, or infinitely many solutions? 3. Can systems of linear equations be solved graphically, and what does the intersection of lines represent? 4. How does the substitution method compare with the elimination method for solving systems of equations? 5. What is the determinant of a matrix, and how can it be used to determine if a system of equations is solvable? **Tip**: For larger systems of equations, using matrices and determinants or a calculator can save time and reduce errors!

Drop file here or Click Here to upload

Algebra
Linear Equations
Systems of Equations

Substitution Method
Elimination Method

Linear Independence Theorem
Gaussian Elimination

Grades 10-12

Solving a System of Linear Equations with Three Variables

Solving a System of Linear Equations Using Elimination

Solving System of Equations with Substitution Method: 7x - 5y + 6z = -3, -3x - 6y = -7, 4x - y - 2z = -5

Solving Systems of Three Linear Equations

Solve a System of Linear Equations: 10x + y + 2z = 8, -x - y + z = 10, and 4x - y + z = 5