Solving a System of 4 Linear Equations Using Substitution

Math Problem Statement

Solve the system of linear equations: 1. x1 + x2 = 1 2. x2 + x3 = 4 3. x3 + x4 = -1 4. x4 = -4

The system of linear equations provided in the image is:

x_1 + x_2 &= 1 \\ x_2 + x_3 &= 4 \\ x_3 + x_4 &= -1 \\ x_4 &= -4 \end{aligned}$$ We can solve this system step by step: 1. From the last equation, we already know: $$x_4 = -4$$ 2. Substituting $$x_4 = -4$$ into the third equation: $$x_3 + (-4) = -1 \implies x_3 = -1 + 4 = 3$$ 3. Now, substituting $$x_3 = 3$$ into the second equation: $$x_2 + 3 = 4 \implies x_2 = 4 - 3 = 1$$ 4. Finally, substituting $$x_2 = 1$$ into the first equation: $$x_1 + 1 = 1 \implies x_1 = 1 - 1 = 0$$ Thus, the solution to the system is: $$x_1 = 0, \quad x_2 = 1, \quad x_3 = 3, \quad x_4 = -4$$ Would you like more details or have any questions? Here are 5 follow-up questions to explore the topic further: 1. How would this solution change if one of the equations was modified? 2. Can you express this system in matrix form and solve it using Gaussian elimination? 3. What are some methods to verify the correctness of this solution? 4. How can you use substitution or elimination methods in systems with more variables? 5. What are the differences between consistent, inconsistent, and dependent systems of equations? **Tip:** Always verify your solution by substituting the values back into the original equations to ensure all conditions are satisfied.

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