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 
 .


We are tasked with finding a vector \( x \) such that \( T(x) = Ax = b \), where \( A \) and \( b \) are given matrices.

### Given:
\[
A = \begin{pmatrix} 
1 & -7 & -10 \\ 
-2 & 5 & 2 
\end{pmatrix}
\]
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.

If T is defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A = [[1, -7, -10], [-2, 5, 2]] and b = [-2, -5].

Explore how to solve the system of linear equations Ax = b, where A is a 2x3 matrix and b is a 2x1 vector. Learn about free variables, matrix multiplication, and the conditions for solution uniqueness. Suitable for undergraduate students studying linear algebra.

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