Suppose that f is a linear function such that $f\left( 3 \right) = 6$and $f\left( { - 2} \right) = 1$. What is $f\left( 8 \right)$?

Hint: In this question it is given that the required equation is a linear equation. Therefore, it should be in the form of $f\left( x \right) = ax + b$. Now, there are two conditions given in the question and we have two variables. So, we can easily find the values of these two variables and then we can find the value of the required function.

Complete step by step answer:
In this question it is given that the required equation is a linear equation. Therefore, we can write it in the form of $ax + b$, where a and b are constants.
Now, first we have two find the value of these constants with the help of two conditions given in the question.
Now,
First condition: $f\left( 3 \right) = 6$
We have $f\left( x \right) = ax + b$ as our equation. Now, put the value of $x = 3$ in the given equation.
$f\left( 3 \right) = 3a + b$
But according to the condition the value of function is $6$.
$3a + b = 6............\left( 1 \right)$
Now,
Second condition: $f\left( { - 2} \right) = 1$
We have $f\left( x \right) = ax + b$ as our equation. Now, put the value of $x = - 2$ in the given equation.
$f\left( { - 2} \right) = - 2a + b$
But according to the condition the value of function is $1$.
$ - 2a + b = 1............\left( 2 \right)$
Now subtract the equation $\left( 2 \right)$ from $\left( 1 \right)$.
We get,
$3a + b - \left( { - 2a + b} \right) = 6 - 1$
On simplification in left-hand side and right-hand side, we get
$ \Rightarrow 3a + 2a = 5$
$ \Rightarrow 5a = 5$
$ \Rightarrow a = 1$
Put this value of a in equation $1$ .
$3\left( 1 \right) + b = 6$
$ \Rightarrow b = 6 - 3$
$ \Rightarrow b = 3$
Therefore, on putting the value of a and b in $f\left( x \right) = ax + b$, we get
 $f\left( x \right) = ax + b$
$ \Rightarrow f\left( x \right) = x + 3$
Now let’s put the value of $x = 8$ in the above equation.
$ \Rightarrow f\left( 8 \right) = 8 + 3$
$ \Rightarrow f\left( 8 \right) = 11$
Therefore, the required value is $11$.

Note:
In this question we have solved two linear equations. In this question we have used an elimination method to solve this question. But we can also solve this question using the method of substitution. In this method we find one variable in terms of another and then we put its value in the second equation.