Question

How do you write a linear function $f$ with the values $f\left(2\right)=-1$ and $f\left(5\right)=4$?

Answer 1

Step-1: Use the linear equation definition to find the two equations:

A linear equation is of the form $f\left(x\right)=ax+b$.

When $x=2$ then $f\left(2\right)=-1$.

Substituting the values in the equation, the equation becomes,

$2a+b=-1...\left(1\right)$

When $x=5$ then $f\left(5\right)=4$.

Substituting the values in the equation, the equation becomes.

$5a+b=4...\left(2\right)$

Step-2: Solve the two equation and find the unknown values:

Subtract equations $\left(1\right)$ and $\left(2\right)$.

$2a+b=-1-5a-b=-4-3a=-5$

Therefore, $a=\frac{5}{3}$.

Now substitute the value $a=\frac{5}{3}$in equation (1)

$$\begin{gathered} 2a+b=-1 \\ \Rightarrow 2\left(\frac{5}{3}\right)+b=-1 \\ \Rightarrow b=-\frac{13}{3} \end{gathered}$$

Step-3: Find the linear equation:

Substitute the values of $a$ and $b$ in the general equation.

$f\left(x\right)=\frac{5}{3}x-\frac{13}{3}$

Hence, the linear equation satisfying the given values is $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{5}}{\mathbf{3}}\mathit{x}\mathbf{-}\frac{\mathbf{13}}{\mathbf{3}}$.