Video: Finding the Value of the Coefficient of π‘₯ in a Linear Function given its Value at a Certain π‘₯-Value

Find the value of π‘˜ given 𝑓(π‘₯) = π‘˜π‘₯ + 13 and 𝑓(8) = βˆ’11.

02:20

Video Transcript

Find the value of π‘˜ given 𝑓 of π‘₯ equals π‘˜π‘₯ plus 13 and 𝑓 of eight is equal to negative 11.

Here is our function. 𝑓 is sort of the name of the function. And π‘₯ is what goes in; it’s its input. We’ve also been given some information about the value of 𝑓 of eight. In other words, our input is no longer π‘₯. It’s eight. And when the input is eight, the output is negative 11. So, let’s see what happens when we put eight into our definition of the function here.

𝑓 of eight means each time we see the π‘₯, we replace or substitute it with eight. So, 𝑓 of eight becomes π‘˜ times eight plus 13. Let’s write that as eight π‘˜ plus 13. But of course, we know that 𝑓 of eight is equal to negative 11. So, this must mean that eight π‘˜ plus 13 must be equal to negative 11. And so, our job, we’re trying find the value of π‘˜, is to solve this equation.

We’ll do this by performing a series of inverse operations. The first thing we’re going to do is subtract 13 from both sides of our equation. Remember, eight π‘˜ plus 13 minus 13 is just eight π‘˜. And negative 11 minus 13 means we move 13 spaces further down the number line. And we get to negative 24. So, eight π‘˜ is equal to negative 24.

The eight is multiplying the π‘˜. And so, the inverse operation we apply next is to divide both sides of our equation by eight. That leaves π‘˜ on the left-hand side. And since 24 divided by eight is three, we know negative 24 divided by eight is negative three. And so, given 𝑓 of π‘₯ is equal to π‘˜π‘₯ plus 13 and 𝑓 of eight is equal to negative 11, we find π‘˜ is equal to negative three.

Now, we can check our answer by letting 𝑓 of π‘₯ now be equal to negative three π‘₯ plus 13. We’ve replaced π‘˜ with negative three. We’re going to check that the value of 𝑓 of eight is indeed negative 11. And so, we replace π‘₯ with eight. And we get negative three times eight plus 13. That’s negative 24 plus 13, which is negative 11 as required.

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