Question

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

30. f(-5) = -4, and f(5) = 2

Answer 1

The required linear equation satisfying the given conditions f(-5)=-4 and f(5)=2 is

`$$y=(3)/(5) x-1$$`

Explanation:

It is given that f(-5)=-4 and f(5)=2. It is required to find out a linear equation satisfying the conditions f(-5)=-4 and f(5)=2. To find it out, first, represent the given conditions in the form of points and then find the slope of a line passing through these two given points. Then consider one of the points to give the 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(-5)=-4 gives the point (5,-4)

And f(5)=2 gives the point (5,2)

This means that the function f(x) satisfies the points (-5,-4) and (5,2).

Step 2 of 4

Now find out the slope of a line passing through the points (-5,-4) and (5,2).

`$$\begin{aligned}m &=(y_(2)-y_(1))/(x_(2)-x_(1)) \\m &=(2-(-4))/(5-(-5)) \\m &=(2+4)/(5+5) \\m &=(6)/(10) \\m &=(3)/(5)\end{aligned}$$`

Step 3 of 4

Now use the slope
`$m=(3)/(5)$` and use one of the two given points and write the equation in point-slope form:

`$$\left(y-y_(2)\right)=m\left(x-x_(2)\right)$$\\ $$(y-2)=(3)/(5)(x-5)$$`

Distribute
`$(3)/(5)$`,

`$$\begin{aligned}&y-2=(3)/(5) x-(3)/(5) * 5 \\&y-2=(3)/(5) x-3\end{aligned}$$`

Step 4 of 4

This linear function can be written in the slope-intercept form by adding 2 on both sides,
`$y-2+2=(3)/(5) x-3+2$`

`$$y=(3)/(5) x-1$$`

So, this is the required linear equation.