Chapter 11.2, Problem 42AYU

In problems 39-74, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

$\{\begin{array}{l}3x-y=7\\ 9x-3y=21\end{array}$

To determine

To find: The solution to the given system of equations using matrices.

Answer to Problem 42AYU

Solution:

Consistent system, infinitely many solutions.

Explanation of Solution

Given:

$3x-y=7$

$9x-3y=21$

Formula used:

To solve a system of two equations in $x$ and $y$ using matrices:

Step 1: Write the corresponding matrix associated with the system of equations.

Step 2: Use elementary row operations to get equivalent matrix of the form:

$\left[\begin{array}{cc}1& a\\ 0& 1\end{array}|\begin{array}{c}b\\ c\end{array}\right]$; where $a\text{, }b\text{, }c$ are constants.

Step 3: Solve for $x$ and $y$.

Calculation:

$3x-y=7$

$9x-3y=21$

The above equations can be rewritten as,

$3x-y=7$

$3\left(3x-y\right)=21$ or $3x-y=7$

We find that both the given equations represent the same equation $3x-y=7$.

Here the number of variables is 2, whereas the number of equation is 1 (less than 2).

Therefore the given system is consistent and has infinitely many solutions.