Math Problem Statement
Solve the system of equations given in Homework-3: Problem 2. The system consists of the following four equations: x1 + x2 = 2, x2 + x3 = 4, x3 + x4 = -1, and x1 + x4 = -3.
Solution
The system of equations in your homework problem can be written as:
x_1 + x_2 &= 2 \\ x_2 + x_3 &= 4 \\ x_3 + x_4 &= -1 \\ x_1 + x_4 &= -3 \end{aligned}$$ We can solve this system step by step. ### Step 1: Solve for $$x_1$$ from the first equation From the first equation: $$x_1 = 2 - x_2$$ ### Step 2: Solve for $$x_3$$ from the second equation From the second equation: $$x_3 = 4 - x_2$$ ### Step 3: Solve for $$x_4$$ from the third equation Substitute the expression for $$x_3$$ into the third equation: $$(4 - x_2) + x_4 = -1$$ $$x_4 = -1 - 4 + x_2 = -5 + x_2$$ ### Step 4: Solve for $$x_2$$ using the fourth equation Substitute the expressions for $$x_1$$ and $$x_4$$ into the fourth equation: $$(2 - x_2) + (-5 + x_2) = -3$$ Simplify: $$2 - x_2 - 5 + x_2 = -3$$ $$-3 = -3$$ This equation is always true, meaning that $$x_2$$ can take any value. Therefore, $$x_2$$ is a free variable. ### Step 5: General solution Since $$x_2$$ is a free variable, we can express the solution in terms of $$x_2$$. Let's set $$x_2 = s$$, where $$s$$ is any real number. Then, the general solution is: $$\begin{aligned} x_1 &= 2 - s \\ x_2 &= s \\ x_3 &= 4 - s \\ x_4 &= -5 + s \end{aligned}$$ Thus, the solution to the system is: $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 2 - s \\ s \\ 4 - s \\ -5 + s \end{bmatrix}$$ This is a parametric solution with $$s$$ as the free parameter. Would you like further details or have any questions? Here are 5 related questions you can explore: 1. How can this system be represented in matrix form? 2. What are the implications of having a free variable in a system of equations? 3. How would the solution change if an additional constraint were applied to the system? 4. How does the system rank help determine whether a system is underdetermined or has a unique solution? 5. What are the possible geometrical interpretations of this solution? **Tip:** When solving systems of linear equations, always check for free variables to see if the system has infinite solutions.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Parametric Solutions
Formulas
Substitute and solve using algebraic methods
General parametric form of solutions for systems of equations
Theorems
Underdetermined system theorem (due to a free variable)
Solution to linear systems with free parameters
Suitable Grade Level
High School - Early College