Chapter 10.1, Problem 31E

To determine

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

  $\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=2\\ \frac{1}{5}x-\frac{2}{3}y=8\end{array}$

Answer to Problem 31E

The given set of equations has one solution $\left(x,y\right)=\left(10,-9\right)$ .

Explanation of Solution

Given:

  $\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=2\\ \frac{1}{5}x-\frac{2}{3}y=8\end{array}$

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:

  $\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=2\\ \frac{1}{5}x-\frac{2}{3}y=8\end{array}$

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

  $\begin{array}{l}y=-\frac{3}{2}x+6\\ y=\frac{3}{10}x-12\end{array}$

Since the slopes are different, the lines must intersect at a point. Here are the graphs of the both eqns:

So, the given set of system has one solution as $\left(x,y\right)=\left(10,-9\right)$ and the graph line is:

  

Conclusion:

Hence, the given set of equations has one solution.