Chapter 6, Problem 22CR

To determine

To solve the system of equations using the method of elimination.

Answer to Problem 22CR

The solution is $x=5,y=12$ & the system of equations has one solution.

Explanation of Solution

Given information:

  $\begin{array}{l}9x+\frac{1}{2}y=51\\ 7x+\frac{1}{3}y=39\end{array}$

Calculation:

  $\begin{array}{l}9x+\frac{1}{2}y=51\to \left(1\right)\\ 7x+\frac{1}{3}y=39\to \left(2\right)\end{array}$

Now,

  $\begin{array}{l}\left[7\times \text{Equation}\left(\text{1}\right)\right]-\left[9\times \text{Equation}\left(\text{2}\right)\right]\\ 7\left(9x+\frac{1}{2}y\right)-9\left(7x+\frac{1}{3}y\right)=7\left(51\right)-9\left(39\right)\\ \Rightarrow 63x+\frac{7}{2}y-63x-3y=357-351\\ \Rightarrow \frac{1}{2}y=6\\ \Rightarrow y=12\end{array}$

Now put $y=12$ in equation (1), we get

  $\begin{array}{l}9x+\frac{1}{2}\left(12\right)=51\\ \Rightarrow 9x+6=51\\ \Rightarrow 9x=45\\ \Rightarrow x=5\end{array}$

Therefore the solution is $x=5,y=12$ & the system of equation has one solution.