Chapter 10.1, Problem 38E

To determine

The given system of equation has to be solved to check the either it has one, no or infinitely number of solutions:

  $\begin{array}{l}2x-3y=-8\\ 14x-21y=3\end{array}$

Answer to Problem 38E

The given set of equations has no solutions.

Explanation of Solution

Given:

  $\begin{array}{l}2x-3y=-8\\ 14x-21y=3\end{array}$

Concept Used:

When the graph line of two equations intersects at a point then one can say that the system of eq. has one solution.

When the graph line of two equations is parallel then one can say that the system of eq. has no solution.

When the graph line of two equations is same then one can say that the system of eq. has infinitely much solution.

The slope intercept form is given as:

  $y=mx+b$

Where,

Slope: $m$

  $y-$ intercept: $b$

Calculations:

The given eqns. are:

  $\begin{array}{l}2x-3y=-8\\ 14x-21y=3\end{array}$

Need to write the given eq. in the slope intercept form:

  $2x-3y=-8$

  $y=\frac{2}{3}x+\frac{8}{3}$.......(1)

And

  $\begin{array}{l}14x-21y=3\\ y=\frac{14}{21}x-\frac{3}{21}\end{array}$

  $y=\frac{2}{3}x-\frac{1}{7}$.......(2)

From eq. (1) and (2):

The set of two eqn. has same slope $\left(m=\frac{2}{3}\right)$ value which clearly represents that lines must be parallel to each other.

The line graph is drawn as below which also shows that these lines are parallel:

  

So, there is no solution.

Conclusion:

Hence, the given set of equations has no solution.