Question

Write a linear function that relates $\mathrm{y}$ to $\mathrm{x}$.

Answer 1

Step 1: Find the slope of the line using the points.

From the above graph Ordered pairs are $(0,5)(1.5,4)(3,3)(4.5,2)$

We know, slope of the line, $\mathrm{m}=\frac{\mathrm{change}\mathrm{in}\mathrm{y}}{\mathrm{change}\mathrm{in}\mathrm{x}}=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}$

Now using two points $(0,5)(1.5,4)$ Here ${\mathrm{x}}_1=0,{\mathrm{x}}_2=1.5$ and ${\mathrm{y}}_1=5,{\mathrm{y}}_2=4$

$$\begin{gathered} \therefore \mathrm{m}=\frac{4-5}{1.5-0} \\ =\frac{-1}{1.5} \\ =-0.67 \end{gathered}$$

Step 2: Finding the linear function that relates $\mathrm{y}$ to $\mathrm{x}$.

We know, Linear function: $\mathrm{y}=\mathrm{mx}+\mathrm{c}$

$\Rightarrow \mathrm{y}=-0.67\mathrm{x}+\mathrm{c}$ $\left[\because \mathrm{m}=1\right]$

$\Rightarrow 5=-0.67\times 0+\mathrm{c}$ (Putting $\mathrm{y}=5$ and $\mathrm{x}=0$ i.e., $(0,5)$)

$$\begin{gathered} \Rightarrow 5=0+c \\ \Rightarrow 5=c \\ \Rightarrow c=5 \end{gathered}$$

Now again put $c=5$ in linear function,

$\Rightarrow \mathrm{y}=-0.67\mathrm{x}+5$

Hence, Linear function would be $\mathrm{y}=-0.67\mathrm{x}+5$.