Math Problem Statement
Solution
To solve each of these systems of linear equations, we'll proceed with the necessary techniques: substitution, elimination, or matrix methods (such as Gaussian elimination). Let's go through each problem step-by-step.
(a) Solve the system:
x + 2y - z = 2 \\ 2x - y + 3z = 9 \\ 3x + 3y - 2z = 3 \end{cases}$$ 1. **Simplify the system** by eliminating one variable, starting with $$z$$. 2. Use substitution or elimination to find $$x$$ and $$y$$. 3. Substitute back to find $$z$$. ### (b) Solve the system: $$\begin{cases} x + y - 10 = 0 \\ x - z - 5 = 0 \\ y - z - 3 = 0 \end{cases}$$ 1. Solve for $$x$$, $$y$$, and $$z$$ by expressing each variable in terms of the others. 2. Substitute accordingly. ### (c) Solve the system: $$\begin{cases} 2x + 4y = 1 \\ 2x + z = 0 \end{cases}$$ 1. Reduce these equations to find the values of $$x$$, $$y$$, and $$z$$. 2. This system may lead to a dependent system or an inconsistency. Verify if there’s a unique solution. ### (d) Solve the system: $$\begin{cases} x + 2y + z + t = 2 \\ x + 3y - z + 2t = 1 \end{cases}$$ 1. Simplify this by eliminating variables using subtraction. 2. Identify if there are dependent or independent variables for potential infinite solutions. ### (e) Solve the system: $$\begin{cases} x + y + z = 2 \\ 3y - z = 1 \\ z = 7 \end{cases}$$ 1. Start with $$z = 7$$, then substitute into the other equations. 2. Solve for $$y$$ and $$x$$. ### (f) Solve the system: $$\begin{cases} x + 2y + z = 0 \\ x + 3y - z = 1 \end{cases}$$ 1. Eliminate $$z$$ by adding or subtracting the equations. 2. Solve for $$x$$ and $$y$$, and then substitute back for $$z$$. --- Would you like the detailed solutions for each part, or do you have questions about a specific system? ### Further Questions: 1. How do you identify if a system has no solution, one solution, or infinitely many solutions? 2. What are the primary methods for solving systems of equations? 3. When is it beneficial to use matrix methods like Gaussian elimination? 4. How can substitution and elimination be combined effectively? 5. What are some common scenarios where systems of linear equations appear in real-world problems? ### Tip: For systems with three or more equations, writing in matrix form and using row reduction can often simplify finding the solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Methods
Formulas
Gaussian elimination
Substitution method
Elimination method
Theorems
Existence and uniqueness theorem for linear systems
Suitable Grade Level
Grades 10-12
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