Chapter 5.3, Problem 19E

To determine

To explain:

An error and correct it by solving the following linear system of equations:

  $\begin{array}{l}5x-7y=16\\ x+7y=8\end{array}$

Answer to Problem 19E

The addition of two equations gives the value of variable $x$ , addition of $5x$ and $x$ should be $6x$ not $4x$ , which gives the wrong value for variable $x$ , so this is the error.

The correct solution of $\begin{array}{l}5x-7y=16\\ x+7y=8\end{array}$ is $x,y=4,\frac{4}{7}$ .

Explanation of Solution

Given information:

  

  $\begin{array}{l}5x-7y=16\\ x+7y=8\end{array}$

Calculation:

The given equations are as:

  $\begin{array}{l}5x-7y=16\\ x+7y=8\end{array}$

The solution is given in the image is:

  $\begin{array}{l}5x-7y=16\\ x+7y=8\\ ---------\\ 4x=24\\ x=6\end{array}$

Now, in the solving method of the given equation, use the elimination method in which here took addition of two equations which results the value of variable $x=6$ .

On observing the given calculation, the error is when two equations are added there is elimination of variable $y$ and addition of $5x$ and $x$ should be $6x$ not $4x$ , which gives the wrong value for variable $x$ .

Now, the correct method of the elimination for given two eqns. is:

  $5x-7y=16$....................... (1)

  $x+7y=8$....................... (2)

Take addition of equation (1) and (2) to eliminate variable $y$ :

  $\begin{array}{l}5x-7y+x+7y=16+8\\ 6x=24\\ x=4\end{array}$

Now calculate the value of variable $y$ by putting value of $x=4$ in eq. (2):

  $\begin{array}{l}x+7y=8\\ 4+7y=8\\ 7y=4\\ y=\frac{4}{7}\end{array}$

Hence, by solving two equations by elimination method get variable values as $x,y=4,\frac{4}{7}$ .

Check for solution:

  $x,y=4,\frac{4}{7}$

Use eq. (2) as:

  $\begin{array}{l}x+7y=8\\ 4+7\left(\frac{4}{7}\right)=8\\ 8=8\end{array}$

Hence, verified.