Chapter 9.4, Problem 30WE

To determine

To find: The value of $a$ .

Answer to Problem 30WE

The value of $a$ is $9$ .

Explanation of Solution

Given:

The given equations are as:

  $\begin{array}{l}5x-3y=35\dots \left(1\right)\\ 7x-3y=43\dots \left(2\right)\\ 4x-ay=61\dots \left(3\right)\end{array}$

The graphs of all the above equations intersect at the same point.

In order to find the value of $a$ , first, it is need to find the intersection point of equations $\left(1\right)$ and $\left(2\right)$ .

So to solve the equation $\left(1\right)$ and $\left(2\right)$ , first, subtract $\left(1\right)$ from equation $\left(2\right)$ ,is as:

  $\begin{array}{c}7x-3y-\left(5x-3y\right)=43-35\\ 2x=8\\ x=4\end{array}$

Further simplified by substituting the value of $x=4$ in equation $\left(1\right)$, as:

  $\begin{array}{c}5\left(4\right)-3y=35\\ -3y=15\\ y=-5\end{array}$

So the intersection point is $\left(4,-5\right)$ .

Now solve the equation $\left(3\right)$ by substituting that above intersecting point at the place of $x$ and $y$ , as any point of the equation satisfied that equation.

  $\begin{array}{c}4\left(4\right)-a\left(-5\right)=61\\ 5a=45\\ a=9\end{array}$

So the third equation after substituting the value of $a$ is given as:

  $4x-9y=61$ .

So the graph of the above given equations is given as:

  

Therefore, the value of $a$ is $9$ .