Math Problem Statement
Solve the following system of equations:
1. x1 - x2 + x3 = 5
2. x1 + 4x2 - x3 = 7
Solution
The system of equations you provided is:
x_1 - x_2 + x_3 = 5 \\ x_1 + 4x_2 - x_3 = 7 \end{cases}$$ This system of linear equations involves three variables $$x_1$$, $$x_2$$, and $$x_3$$. Since there are only two equations, this is an underdetermined system, meaning it could have infinitely many solutions or no solution depending on the consistency of the equations. ### Solving the System To solve this system, we can express it in matrix form and use row operations, or simply solve it by substitution or elimination. 1. **Add the two equations:** $$(x_1 - x_2 + x_3) + (x_1 + 4x_2 - x_3) = 5 + 7$$ Simplifying: $$2x_1 + 3x_2 = 12 \quad \text{(Equation 3)}$$ 2. **Solve for one variable:** From Equation 3: $$x_1 = \frac{12 - 3x_2}{2} = 6 - \frac{3}{2}x_2$$ 3. **Substitute back into one of the original equations to solve for $$x_3$$:** Using the first equation: $$(6 - \frac{3}{2}x_2) - x_2 + x_3 = 5$$ Simplify and solve for $$x_3$$: $$6 - \frac{5}{2}x_2 + x_3 = 5$$ $$x_3 = -1 + \frac{5}{2}x_2$$ ### General Solution The general solution is therefore: $$x_1 = 6 - \frac{3}{2}x_2$$ $$x_3 = -1 + \frac{5}{2}x_2$$ Here, $$x_2$$ can be any real number, giving us an infinite number of solutions depending on the value of $$x_2$$. Would you like more details or have any specific questions? --- **Related Questions:** 1. What is the significance of having more variables than equations in a system? 2. How can this solution be interpreted geometrically? 3. What are the conditions for a system of linear equations to have a unique solution? 4. How would the solution change if another equation were added to the system? 5. How do you determine if a system is consistent or inconsistent? **Tip:** When dealing with underdetermined systems, always check if the system is consistent before attempting to solve it.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Formulas
Substitution Method
Elimination Method
Theorems
Theorem of Underdetermined Systems
Suitable Grade Level
Grades 10-12
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