Question

31. For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.

Answer 1

Answer:

The required linear equation satisfying the given conditions f(-1)=4 and f(5)=1 is $y=\frac{-1}{2} x+\frac{7}{2}$

Step-by-step explanation:

It is given that f(-1)=4 and f(5)=1.

It is required to find out a linear equation satisfying the conditions f(-1)=4

and f(5)=1. linear equation of the line in the form

$\left(y-y_{2}\right)=m\left(x-x_{2}\right)$

Step 1 of 4

Observe, f(-1)=4 gives the point (-1,4)

And f(5)=1 gives the point (5,1).

This means that the function f(x) satisfies the points (-1,4) and (5,1).

Step 2 of 4

Now find out the slope of a line passing through the points (-1,4) and (5,1),

$\begin{aligned}&m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\&m=\frac{1-4}{5-(-1)} \\&m=\frac{-3}{5+1} \\&m=\frac{-3}{6} \\&m=\frac{-1}{2}\end{aligned}$

Step 3 of 4

Now use the slope $m=\frac{-1}{2}$ and use one of the two given points and write the equation in point-slope form:

$(y-1)=\frac{-1}{2}(x-5)$

Distribute $\frac{-1}{2}$,

$y-1=\frac{-1}{2} x+\frac{5}{2}$

Step 4 of 4

This linear function can be written in the slope-intercept form by adding 1 on both sides,

$\begin{aligned}&y-1+1=\frac{-1}{2} x+\frac{5}{2}+1 \\&y=\frac{-1}{2} x+\frac{5}{2}+\frac{2}{2} \\&y=\frac{-1}{2} x+\frac{7}{2}\end{aligned}$

So, this is the required linear equation.