Write a linear function $f$ with $f\left(5\right)=-1$ and $f\left(0\right)=-5$ .

Question

Can you please solve this for me. I am so confused right now...

Answer 1

I'm going to rewrite the question to improve syntax understanding to clarify what I am doing.

Write a linear function, f, with f(5) = -1 and f(0) = -5.

When we have f(x), we know it we are going to have a function of x, meaning a function with x being the independent variable.

Here we are given two functions:

f(5) = -1, when x is equal to 5, y is equal to -1

similarly,

f(0) = -5, when x is equal to 0, y is equal to -5

We want to find a linear function that makes both of the above, given statements true.

A linear function is of the form y= mx+b.

There are two ways to solve this type of problem. A system of equations, or the specific equations derived from the system of equations.

The latter is the following process:

m = (y1-y0)/(x1-x0) either x and y can be the y1, y0 or x1, x0 as long as the proper y1 and x1 correspond

here we have

m = (-1 -(-5))/(5-0) = (-1+5)/(5-0) = 4/5

now we can plug this into y=mx+b

plug in m, and corresponding y and x

it will work with either pair of y and x

-1 = (4/5)*(5) + b Solve for b

-1 = (20/5) + b

-1 = 4 +b subtract 4 from both sides

-5 = b

the linear equation is the y = (4/5)*x -5

we can also solve using the other pair of (x,y) coordinates

-5 = (4/5)*(0) + b

-5 = b

as you can see here this is a much simpler process because x = 0.

If we were to solve this as a system of equations we would have

-1 = (m*5) +b AND -5 = (m*0) + b

this solves easy here as b = -5

we can then plug b into the other equation and solve for m.

Two equations and two unknowns yields the linear equation.