Math Problem Statement
Solve the system of linear equations:
1. x1 + x2 = 1
2. x2 + x3 = 4
3. x3 + x4 = -4
4. x1 + x4 = -7
Solution
The system of linear equations given in the image is:
x_1 + x_2 &= 1, \\ x_2 + x_3 &= 4, \\ x_3 + x_4 &= -4, \\ x_1 + x_4 &= -7. \end{aligned}$$ To solve this system, we can use substitution or elimination methods. Let’s proceed step by step: ### Step 1: Express $$x_1$$ in terms of $$x_2$$ from the first equation. From the first equation $$x_1 + x_2 = 1$$, we solve for $$x_1$$: $$x_1 = 1 - x_2.$$ ### Step 2: Substitute $$x_1 = 1 - x_2$$ into the fourth equation. The fourth equation is $$x_1 + x_4 = -7$$. Substituting $$x_1 = 1 - x_2$$ into this equation: $$(1 - x_2) + x_4 = -7,$$ which simplifies to: $$x_4 = -8 + x_2.$$ ### Step 3: Substitute $$x_4 = -8 + x_2$$ into the third equation. The third equation is $$x_3 + x_4 = -4$$. Substituting $$x_4 = -8 + x_2$$: $$x_3 + (-8 + x_2) = -4,$$ which simplifies to: $$x_3 = 4 - x_2.$$ ### Step 4: Substitute $$x_3 = 4 - x_2$$ into the second equation. The second equation is $$x_2 + x_3 = 4$$. Substituting $$x_3 = 4 - x_2$$: $$x_2 + (4 - x_2) = 4,$$ which simplifies to: $$4 = 4.$$ This equation is always true, meaning that $$x_2$$ is a free variable. ### Final Solution: Since $$x_2$$ is a free variable, let $$x_2 = s$$, where $$s$$ is any real number. Now express the other variables in terms of $$s$$: $$x_1 = 1 - s,$$ $$x_3 = 4 - s,$$ $$x_4 = -8 + s.$$ Thus, the solution to the system is: $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ -4 \\ -7 \end{bmatrix} + s \begin{bmatrix} -1 \\ 1 \\ -1 \\ 1 \end{bmatrix},$$ where $$s$$ is a free parameter. Would you like more details or have any questions? ### Related questions: 1. How does Gaussian elimination work for solving systems of linear equations? 2. Can systems of linear equations have no solution or infinitely many solutions? 3. What is the significance of free variables in solving systems of equations? 4. How can matrix notation help in solving systems of linear equations? 5. How does the row-reduction method compare to substitution for solving linear systems? ### Tip: In systems of linear equations, if an equation simplifies to something always true (like $$4 = 4$$), it usually means that one or more variables are free, allowing for infinitely many solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Representation
Formulas
x1 + x2 = 1
x2 + x3 = 4
x3 + x4 = -4
x1 + x4 = -7
Theorems
Substitution method
Gaussian Elimination
Parametric solution
Suitable Grade Level
Undergraduate (Linear Algebra)