Math Problem Statement
If T is defined by T(x)equalsAx, find a vector x whose image under T is b, and determine whether x is unique. Let AequalsStart 2 By 3 Table 1st Row 1st Column 1 2nd Column negative 7 3rd Column negative 10 2nd Row 1st Column negative 2 2nd Column 5 3rd Column 2 EndTable and bequalsStart 2 By 1 Table 1st Row 1st Column negative 2 2nd Row 1st Column negative 5 EndTable .
Solution
We are tasked with finding a vector such that , where and are given matrices.
Given:
1 & -7 & -10 \\ -2 & 5 & 2 \end{pmatrix}$$ and $$b = \begin{pmatrix} -2 \\ -5 \end{pmatrix}$$ We need to solve for $$x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$$ such that: $$A \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} -2 \\ -5 \end{pmatrix}$$ This corresponds to solving the system of linear equations: 1. $$1x_1 - 7x_2 - 10x_3 = -2$$ 2. $$-2x_1 + 5x_2 + 2x_3 = -5$$ ### Step 1: Solve the System of Equations Let's solve this system using a method such as substitution or elimination. $$\begin{aligned} & 1x_1 - 7x_2 - 10x_3 = -2 \quad \text{(Equation 1)} \\ & -2x_1 + 5x_2 + 2x_3 = -5 \quad \text{(Equation 2)} \end{aligned}$$ I'll solve this system to find $$x_1, x_2, x_3$$.The solution to the system is: $$x_1 = 5 - 4x_3$$ $$x_2 = 1 - 2x_3$$ Since $$x_3$$ is a free variable, the solution for $$x$$ is not unique. The general solution is: $$x = \begin{pmatrix} 5 - 4x_3 \\ 1 - 2x_3 \\ x_3 \end{pmatrix}$$ This means there are infinitely many solutions, parameterized by the value of $$x_3$$. Would you like a more detailed breakdown or further clarification? Here are 5 questions you could explore next: 1. How do free variables affect the uniqueness of solutions? 2. What is the geometric interpretation of a system with infinitely many solutions? 3. Can you determine specific solutions by choosing values for $$x_3$$? 4. What conditions guarantee a unique solution for a system of linear equations? 5. How can this method be extended to larger systems of equations? **Tip:** For systems of equations, checking the rank of the matrix can help identify if a solution is unique, non-existent, or infinite.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Multiplication
Formulas
Ax = b
Theorems
System of Linear Equations Solution
Free Variable Theorem
Suitable Grade Level
Undergraduate Mathematics
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