Problem 33 - Elementary and Intermediate Algebra

Question

If the systems corresponding to the matrices \(\left[\begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {d_{1}} & {e_{1}} & {f_{1}}\end{array}\right]\) and \(\left[\begin{array}{lll}{a_{2}} & {b_{2}} & {c_{2}} \\ {d_{2}} & {e_{2}} & {f_{2}}\end{array}\right]\) share the same solution, does it follow that the corresponding entries are all equal to each other \(\left(a_{1}=a_{2}, b_{1}=b_{2}, \text { etc. }\right) ?\) Why or why not?

Step-by-Step Solution

Answer

No, the entries do not have to be equal for the systems to share the same solution.

Step 1: Understand the Problem

We need to determine if the two systems of equations represented by the matrices \(\begin{bmatrix} a_{1} & b_{1} & c_{1} \ d_{1} & e_{1} & f_{1} d{bmatrix}\) and \(\begin{bmatrix} a_{2} & b_{2} & c_{2} \ d_{2} & e_{2} & f_{2} d{bmatrix}\) have the same entries if they share the same solution.

Step 1: Write the Systems of Equations

The first matrix represents the system of equations:\(a_{1}x + b_{1}y = c_{1}d_{1}x + e_{1}y = f_{1}\)The second matrix represents the system of equations:\(a_{2}x + b_{2}y = c_{2}d_{2}x + e_{2}y = f_{2}\)

Step 2: Consider Same Solutions

Assume both systems have the same solution \( (x,y) \).So, \(x = x_0\) and \(y = y_0\). Substitute these values into both systems:\(a_{1}x_0 + b_{1}y_0 = c_{1}\)\(d_{1}x_0 + e_{1}y_0 = f_{1}\)and\(a_{2}x_0 + b_{2}y_0 = c_{2}\)\(d_{2}x_0 + e_{2}y_0 = f_{2}\)

Step 3: Compare Coefficients

Having \( (x_0, y_0) \ ) as a solution means:\(a_{1}x_0 + b_{1}y_0 = a_{2}x_0 + b_{2}y_0 \) and \( c_{1} = c_{2} \), d_{1}x_0 + e_{1}y_0 = d_{2}x_0 + e_{2}y_0\) and \( f_{1} = f_{2} \). But these conditions do not require equal coefficients. Different coefficients can still sum to the same result.

Step 4: Provide a Counterexample

Consider the system \(\begin{bmatrix} 1 & 1 & 2 \ 1 & 2 & 3 d{bmatrix}\) and \(\begin{bmatrix} 2 & 2 & 4 \ 2 & 4 & 6 d{bmatrix}\). The solution for both is \( (1, 1)\) even though the entries are not all equal. This shows different matrices can have the same solutions without having identical entries.

Key Concepts

matrices

In algebra, matrices are rectangular arrays of numbers arranged in rows and columns. Each element in the matrix has a specific position. They are used to represent systems of linear equations compactly. For example, the matrices \(\begin{bmatrix} a_1 & b_1 & c_1 \ d_1 & e_1 & f_1 \end{bmatrix}\) and \(\begin{bmatrix} a_2 & b_2 & c_2 \ d_2 & e_2 & f_2 \end{bmatrix}\) represent sets of linear equations. The entries \(a_1, b_1, c_1\) form the first row and correspond to the coefficients of the variables in one equation. Similarly, \(d_1, e_1, f_1\) form the second row.
Matrices are a concise way to describe and manipulate systems of equations, making calculations more efficient.

solutions

The solution to a system of linear equations is a set of values for the variables that satisfy all equations simultaneously. For instance, for the systems discussed, if \( (x_0, y_0) \) solves the first system, it must satisfy the equations derived from the matrix entries, meaning: \[ a_1 x_0 + b_1 y_0 = c_1 \ d_1 x_0 + e_1 y_0 = f_1 \] Likewise, the same \((x_0, y_0)\) would satisfy the second system represented by \[ a_2 x_0 + b_2 y_0 = c_2 \ d_2 x_0 + e_2 y_0 = f_2 \] Note that though both systems might have the same solution, it does not imply that their respective entries must be identical.

coefficients

Coefficients are the numerical or symbolic multipliers of variables in equations. In the given systems, \(a_1, b_1\) and \(a_2, b_2\) are coefficients of \(x\) and \(y\) in the first equations, respectively. These coefficients affect the slope and position of the lines represented by the equations. Even if two systems share the same solution \((x_0, y_0)\), their coefficients \(a_i\) and \(b_i\) can differ. It is the combination and interaction of these coefficients with the variables that determine the resulting values, not their equality.

counterexample

A counterexample disproves a statement by providing a specific case where the statement fails. In the given exercise, a counterexample helps demonstrate that the assumption that identical solutions imply identical coefficients is incorrect. Consider the matrices \[\begin{bmatrix} 1 & 1 & 2 \ 1 & 2 & 3 \end{bmatrix} \ and \begin{bmatrix} 2 & 2 & 4 \ 2 & 4 & 6 \end{bmatrix} \]. Both systems share the solution \((1,1)\) despite having different coefficients. This shows that having the same solution does not necessitate equal entries in the matrices. The counterexample clearly confirms the independence of solutions from the equality of coefficients.