Math Problem Statement

Tìm hệ nghiệm cơ bản của hệ phương trình tuyến tính sau: x1 + x2 - x3 + x4 = 0, x1 - 3x3 = 0.

Solution

To solve the given system of linear equations:

1. & \quad x_1 + x_2 - x_3 + x_4 = 0, \\ 2. & \quad x_1 - 3x_3 = 0, \end{aligned}$$ we need to find the basic solutions for the system. ### Step 1: Express the variables in terms of free variables From the second equation, solve for $$x_1$$: $$x_1 = 3x_3.$$ Substitute $$x_1 = 3x_3$$ into the first equation: $$3x_3 + x_2 - x_3 + x_4 = 0.$$ Simplify: $$2x_3 + x_2 + x_4 = 0.$$ Rearrange to solve for $$x_2$$: $$x_2 = -2x_3 - x_4.$$ ### Step 2: Define free variables We let $$x_3$$ and $$x_4$$ be the free variables. Define: $$x_3 = t, \quad x_4 = s,$$ where $$t$$ and $$s$$ are parameters. ### Step 3: Express all variables Using the above relationships: $$x_1 = 3x_3 = 3t,$$ $$x_2 = -2x_3 - x_4 = -2t - s.$$ Thus, the general solution can be written as: $$(x_1, x_2, x_3, x_4) = (3t, -2t - s, t, s).$$ ### Step 4: Write the solution in vector form Factorize the parameters $$t$$ and $$s$$: $$(x_1, x_2, x_3, x_4) = t(3, -2, 1, 0) + s(0, -1, 0, 1).$$ ### Basic solution The basic solutions of the system are: $$(3, -2, 1, 0) \quad \text{and} \quad (0, -1, 0, 1).$$ Let me know if you have further questions or want additional explanations! --- ### Follow-up Questions: 1. What does the concept of free variables mean in a system of equations? 2. How is a basic solution set derived from a parametric representation? 3. What is the geometric interpretation of the solutions in a four-dimensional space? 4. How would the solution change if an additional equation was added to the system? 5. Can the rank of the coefficient matrix be used to determine the number of basic solutions? ### Tip: When solving systems of linear equations, always simplify the equations to express as many variables as possible in terms of free variables to make the general solution easier to interpret.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Parametric Solutions

Formulas

General form of a linear equation: ax1 + bx2 + cx3 + ... = d
Parametric representation: x = x_free * vector

Theorems

Gaussian elimination
Basic solutions in vector spaces

Suitable Grade Level

Undergraduate (Linear Algebra) or Advanced High School

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