Chapter 6.CT, Problem 1CT

To determine

To find:

The solution of the system of equation $3x+2y=1$ and $2x-y=-11$ by any method and to check if the system is consistent or inconsistent and to also check whether the system is dependent or independent

Answer to Problem 1CT

Solution:

The solution of the system of equation is $\left(-3,5\right)$ and the system is consistent and the equations are independent.

Explanation of Solution

Given:

The system of equation is $3x+2y=1$ and $2x-y=-11$.

The system of equation is,

$3x+2y=1\cdots (1)$

And,

$2x-y=-11\cdots (2)$

Multiply the equation $\left(2\right)$ by $2$ and add to equation $\left(1\right)$.

$\begin{array}{l}3x+2y=1\\ 4x-2y=-22\\ \underset{\_}{+++}\\ 7x=-21\end{array}$

Further solve the above equation.

$\begin{array}{rll}x& =& -\frac{21}{7}\\ x& =& -3\end{array}$

Substitute $-3$ for $x$ in equation $\left(1\right)$.

$\begin{array}{rll}3\left(-3\right)+2y& =& 1\\ 2y& =& 1+9\\ 2y& =& 10\\ y& =& 5\end{array}$

There is only single solution of $x$ and $y$. The system of equation preserves only one solution therefore, it is consistent and the equations are independent.

Conclusion:

Thus, the solution of the system of equation is $\left(-3,5\right)$ and the system is consistent and the equations are independent.