Math Problem Statement
Solution
The task asks us to verify that the given parametric vector is a solution to the given system of linear equations for all values of . Let's break this down step by step.
Given:
The vector is: The system of linear equations is:
x_1 + x_2 + x_3 &= 6 \\ x_1 - x_2 - 2x_3 &= -7 \\ 5x_1 + x_2 - x_3 &= 4 \end{aligned}$$ ### Substituting the parametric expressions for $$x_1$$, $$x_2$$, and $$x_3$$: 1. **First equation**: $$x_1 + x_2 + x_3 = 6$$ Substituting $$x_1 = 1 - t$$, $$x_2 = 2 + 3t$$, and $$x_3 = 3 - 2t$$: $$(1 - t) + (2 + 3t) + (3 - 2t) = 6$$ Simplifying: $$1 - t + 2 + 3t + 3 - 2t = 6$$ $$6 + 0t = 6$$ This is true for all values of $$t$$. 2. **Second equation**: $$x_1 - x_2 - 2x_3 = -7$$ Substituting $$x_1 = 1 - t$$, $$x_2 = 2 + 3t$$, and $$x_3 = 3 - 2t$$: $$(1 - t) - (2 + 3t) - 2(3 - 2t) = -7$$ Simplifying: $$1 - t - 2 - 3t - 6 + 4t = -7$$ $$-7 + 0t = -7$$ This is also true for all values of $$t$$. 3. **Third equation**: $$5x_1 + x_2 - x_3 = 4$$ Substituting $$x_1 = 1 - t$$, $$x_2 = 2 + 3t$$, and $$x_3 = 3 - 2t$$: $$5(1 - t) + (2 + 3t) - (3 - 2t) = 4$$ Simplifying: $$5 - 5t + 2 + 3t - 3 + 2t = 4$$ $$4 + 0t = 4$$ This is true for all values of $$t$$. ### Conclusion: The vector $$(1 - t, 2 + 3t, 3 - 2t)$$ satisfies all three equations for all values of $$t$$, hence it is a solution to the system for any $$t$$. Would you like more details or have any questions? Here are 5 related questions to expand on this: 1. What is the general method to solve systems of linear equations using substitution? 2. How can we interpret parametric solutions geometrically? 3. What are some other methods to solve linear systems besides substitution? 4. Can parametric solutions apply to non-linear systems? 5. How does Gaussian elimination work in solving systems of equations? **Tip:** When dealing with systems of linear equations, always check if the number of equations matches the number of variables to ensure a well-posed problem!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 Linear Equations
Parametric Equations
Formulas
x1 + x2 + x3 = 6
x1 - x2 - 2x3 = -7
5x1 + x2 - x3 = 4
Theorems
Solution verification for systems of equations
Substitution method in algebra
Suitable Grade Level
Grades 10-12 or early university
Related Recommendation
Determine Equivalence of Parametric Solution for a Three-Variable System
Solving Parametric Linear Equations: x1 = t, x2 = 2 + t, x3 = 3 - t
Solve Parametric Equations y = 3t - t^2, x = 3t^2 at t = 1
Determine if (9, 2, 1) is a Solution for a System of Dependent Equations
Eliminating Parameters in Parametric Equations: y = 3t - t^2, x = 3t^2