# Question Video: Finding the Value of the Coefficient of 𝑥 in a Linear Function given its Value at a Certain 𝑥-Value Mathematics • 8th Grade

Find the value of 𝑘 given 𝑓(𝑥) = 𝑘𝑥 + 13 and 𝑓(8) = −11.

02:20

### Video Transcript

Find the value of 𝑘 given 𝑓 of 𝑥 equals 𝑘𝑥 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 𝑥 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 𝑥. 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 𝑥, we replace or substitute it with eight. So, 𝑓 of eight becomes 𝑘 times eight plus 13. Let’s write that as eight 𝑘 plus 13. But of course, we know that 𝑓 of eight is equal to negative 11. So, this must mean that eight 𝑘 plus 13 must be equal to negative 11. And so, our job, we’re trying find the value of 𝑘, 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 𝑘 plus 13 minus 13 is just eight 𝑘. And negative 11 minus 13 means we move 13 spaces further down the number line. And we get to negative 24. So, eight 𝑘 is equal to negative 24.

The eight is multiplying the 𝑘. And so, the inverse operation we apply next is to divide both sides of our equation by eight. That leaves 𝑘 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 𝑥 is equal to 𝑘𝑥 plus 13 and 𝑓 of eight is equal to negative 11, we find 𝑘 is equal to negative three.

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