Math Problem Statement
Solve the system of equations: x1 + x2 = -1, x2 + x3 = 2, x3 + x4 = 4, x1 + x4 = 1.
Solution
The given system of equations is:
x_1 + x_2 &= -1 \\ x_2 + x_3 &= 2 \\ x_3 + x_4 &= 4 \\ x_1 + x_4 &= 1 \end{aligned}$$ We need to solve this system for $$x_1, x_2, x_3, x_4$$. ### Step 1: Solve for one variable in terms of others Let's begin by solving the first equation for $$x_1$$: $$x_1 = -1 - x_2$$ ### Step 2: Substitute into the fourth equation We can substitute $$x_1 = -1 - x_2$$ into the fourth equation: $$(-1 - x_2) + x_4 = 1$$ $$x_4 = 1 + 1 + x_2 = 2 + x_2$$ Now we have expressions for $$x_1$$ and $$x_4$$ in terms of $$x_2$$. ### Step 3: Use the second and third equations Next, substitute these into the second and third equations: 1. The second equation is $$x_2 + x_3 = 2$$. Solve for $$x_3$$: $$x_3 = 2 - x_2$$ 2. The third equation is $$x_3 + x_4 = 4$$. Substitute $$x_3 = 2 - x_2$$ and $$x_4 = 2 + x_2$$ into this: $$(2 - x_2) + (2 + x_2) = 4$$ This simplifies to: $$4 = 4$$ This equation is always true, so it doesn't provide new information. ### Step 4: Conclusion From the analysis, we can express all variables in terms of $$x_2$$: $$\begin{aligned} x_1 &= -1 - x_2 \\ x_2 &= x_2 \\ x_3 &= 2 - x_2 \\ x_4 &= 2 + x_2 \end{aligned}$$ Thus, the solution to the system is: $$\mathbf{x} = \begin{pmatrix} -1 - s \\ s \\ 2 - s \\ 2 + s \end{pmatrix}$$ where $$s$$ is any real number (a free parameter). Would you like more details on the steps or further clarifications? ### Related Questions: 1. How do you check if a system has infinitely many solutions? 2. What are the conditions for a system to have a unique solution? 3. How can you interpret the free parameter $$s$$ geometrically? 4. What are different methods to solve systems of linear equations? 5. How does this system relate to matrix representation of linear equations? ### Tip: When dealing with systems of equations, it's useful to reduce the system to one or two variables by substitution or elimination to simplify the solution process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Substitution Method
Free Parameter
Formulas
x1 = -1 - x2
x3 = 2 - x2
x4 = 2 + x2
Theorems
System of Linear Equations Theorem
Suitable Grade Level
Grades 9-12