Simplify and solve the following linear equations.
$3\left( 5z-7 \right)-2\left( 9z-11 \right)=4\left( 8z-13 \right)-17$
A $z=2$
B $z=9$
C $z=3$
D $z=7$

Hint: To obtain the solution of the given equation use simple addition and subtraction by opening the brackets. Firstly we open the bracket by multiplying the term inside it by the term outside it. Then we will simplify the equation on both sides. Finally we will take the term having variable $z$ on one side and solve the equation to get the desired answer.

Complete step-by-step solution:
The linear equation is given as below:
 $3\left( 5z-7 \right)-2\left( 9z-11 \right)=4\left( 8z-13 \right)-17$
Firstly we will open the bracket on both sides and simplify it as follows:
$\begin{align}
  & \Rightarrow 3\times \left( 5z \right)-3\times \left( 7 \right)-2\times \left( 9z \right)-2\times \left( -11 \right)=4\times \left( 8z \right)+4\times \left( -13 \right)-17 \\
 & \Rightarrow 15z-21-18z+22=32z-52-17 \\
 & \Rightarrow 1-3z=32z-69 \\
\end{align}$
Next we will take all the term having variable $z$ on left side of the equation and the rest term on the right side as follows:
$\begin{align}
  & \Rightarrow -3z-32z=-1-69 \\
 & \Rightarrow -35z=-70 \\
\end{align}$
Now we will divide both sides by -35 as follows:
$\begin{align}
  & \Rightarrow \dfrac{-35z}{-35}=\dfrac{-70}{-35} \\
 & \therefore z=2 \\
\end{align}$
So the value is obtained as $z=2$
Hence option (A) is correct.

Note: Linear equations are those equations which have a variable with order one or we can say that the equation that is of first order is known as Linear Equation. If the equations have only one variable it is known as the Linear equation in one variable. We can have more than one variable also but there degree should always be 1 for it to be a Linear equation. The solutions of the Linear equation are the values that satisfy the equation completely. We can check whether the solution obtained is right or not by substituting it in the equation as follows:
Put $z=2$ in $3\left( 5z-7 \right)-2\left( 9z-11 \right)=4\left( 8z-13 \right)-17$
$\begin{align}
  & \Rightarrow 3\left( 5\times 2-7 \right)-2\left( 9\times 2-11 \right)=4\left( 8\times 2-13 \right)-17 \\
 & \Rightarrow 3\left( 10-7 \right)-2\left( 18-11 \right)=4\left( 16-13 \right)-17 \\
 & \Rightarrow 3\times 3-2\times 7=4\times 3-17 \\
 & \Rightarrow -5=-5 \\
\end{align}$
We got the right side equal to the left side hence our solution is correct.