If 𝑓 of 𝑥 is a linear function, find an equation for it given 𝑓 of four equals negative one and 𝑓 of nine equals two.
To help us solve this problem, what we’ve got is the general form for a linear function. And that general form is that 𝑦 or 𝑓 of 𝑥 is equal to 𝑚𝑥 plus 𝑐, where 𝑚 is the slope and 𝑐 is the 𝑦-intercept. So, therefore, we know that our linear function is gonna take this form. It’s worth, at this point, reminding ourselves that 𝑓 of 𝑥 is 𝑦 or it’s another way that we could write what the function is. So what we’re going to do is use our values that we know, trying to form a couple of simultaneous equations.
First of all, we have 𝑓 of four equals negative one. What this means is that the value of the function is negative one when 𝑥 is equal to four. So we can say that four 𝑚, and that’s because our 𝑥-value is four, plus 𝑐 is gonna be equal to negative one cause all I’ve done is flipped the equation round the other way. And I’m gonna call this equation equation one. And the reason I’m labelling the equation is cause I find it useful when we go through the next steps.
Next, we know that 𝑓 of nine is equal to two. So again, this tells us that the value of the function is two when the value of 𝑥 equals nine. So we now have our second simultaneous equation. And that is that nine 𝑚 plus 𝑐 is equal to two. And I’ve labelled this equation two. So now, what we’re gonna do is use a method called elimination to solve our simultaneous equations. So to use elimination, all we need to do is find terms, so our variables, that have the same coefficient. So if you look at these, the 𝑚s, well, they don’t have the same coefficient cause they got four and nine. However, the 𝑐s do because the coefficient of our 𝑐s is just one. But what do we do with them once we found this out?
Well then, we’re either gonna add or subtract our equations from each other. And here we can see that we’ve got the same signs because both 𝑐s are positive. And as the signs are the same, what we’re gonna do is subtract. We can say that same sign subtract. And that’s because we want to eliminate the different values, so eliminate 𝑐 from the equation. So we can solve to find 𝑚. And if you have positive 𝑐 minus positive 𝑐, then we’re just gonna get zero.
So now, what we’re going to do is equation two minus equation one. So we’re going to get five 𝑚. And that’s because nine 𝑚 minus four 𝑚 is five 𝑚. And then the 𝑐s have cancelled out cause, as we said, positive 𝑐 minus positive 𝑐 is zero. And then on the right-hand side of the equation, we’re gonna have two minus negative one. Well, if you subtract a negative number, it’s the same as adding that number on. So we have two add one which is three.
So now, we’re gonna solve to find 𝑚. And to do that, what we need to do is divide each side of the equation by five. And when we do that, we get 𝑚 is equal to three over five or three-fifths. So we’ve now found the slope of our linear function. So now, what we want to do is find 𝑐, our 𝑦 intercept. But how we’re going to do that?
Well, now to enable us to find 𝑐, what we’re going to do is substitute 𝑚 equals three-fifths into one of our equations. I’ve chosen equation one. But it could be either of them. So when we substitute 𝑚 equals three-fifths into equation one, we get four multiplied by three-fifths plus 𝑐 is equal to negative one which is gonna give us twelve-fifths because we multiplied four by the numerator. So we’ve got twelve-fifths plus 𝑐 is equal to negative one. So now, if we subtract twelve-fifths from both sides of the equation, we’re gonna get 𝑐 is equal to negative seventeen-fifths. And we got that as a value for 𝑐 because we had negative one minus twelve-fifths. Well, we can think of one as five-fifths. So, therefore, we’d have negative five-fifths minus twelve-fifths which gives us negative seventeen-fifths.
So therefore if we substitute our values of 𝑚 and 𝑐, which are 𝑚 is equal to three-fifths and 𝑐 is equal to negative seventeen-fifths, into the linear function general form, then we’ll get 𝑓 of 𝑥, because we are using the function notation, is equal to negative seventeen-fifths plus three-fifths 𝑥. Or it can be written as 𝑓 of 𝑥 is equal to three-fifths 𝑥 minus seventeen-fifths.