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If π of π₯ is a linear function, find an equation for it given π of four equals negative one and π of nine equals two.
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To help us solve this problem, what weβve got is the general form for a linear function.
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And that general form is that π¦ or π of π₯ is equal to ππ₯ plus π, where π is the slope and π is the π¦-intercept.
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So, therefore, we know that our linear function is gonna take this form.
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Itβs worth, at this point, reminding ourselves that π of π₯ is π¦ or itβs another way that we could write what the function is.
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So what weβre going to do is use our values that we know, trying to form a couple of simultaneous equations.
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First of all, we have π of four equals negative one.
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What this means is that the value of the function is negative one when π₯ is equal to four.
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So we can say that four π, and thatβs because our π₯-value is four, plus π is gonna be equal to negative one cause all Iβve done is flipped the equation round the other way.
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And Iβm gonna call this equation equation one.
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And the reason Iβm labelling the equation is cause I find it useful when we go through the next steps.
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Next, we know that π of nine is equal to two.
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So again, this tells us that the value of the function is two when the value of π₯ equals nine.
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So we now have our second simultaneous equation.
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And that is that nine π plus π is equal to two.
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And Iβve labelled this equation two.
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So now, what weβre gonna do is use a method called elimination to solve our simultaneous equations.
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So to use elimination, all we need to do is find terms, so our variables, that have the same coefficient.
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So if you look at these, the πs, well, they donβt have the same coefficient cause they got four and nine.
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However, the πs do because the coefficient of our πs is just one.
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But what do we do with them once we found this out?
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Well then, weβre either gonna add or subtract our equations from each other.
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And here we can see that weβve got the same signs because both πs are positive.
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And as the signs are the same, what weβre gonna do is subtract.
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We can say that same sign subtract.
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And thatβs because we want to eliminate the different values, so eliminate π from the equation.
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So we can solve to find π.
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And if you have positive π minus positive π, then weβre just gonna get zero.
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So now, what weβre going to do is equation two minus equation one.
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So weβre going to get five π.
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And thatβs because nine π minus four π is five π.
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And then the πs have cancelled out cause, as we said, positive π minus positive π is zero.
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And then on the right-hand side of the equation, weβre gonna have two minus negative one.
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Well, if you subtract a negative number, itβs the same as adding that number on.
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So we have two add one which is three.
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So now, weβre gonna solve to find π.
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And to do that, what we need to do is divide each side of the equation by five.
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And when we do that, we get π is equal to three over five or three-fifths.
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So weβve now found the slope of our linear function.
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So now, what we want to do is find π, our π¦ intercept.
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But how weβre going to do that?
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Well, now to enable us to find π, what weβre going to do is substitute π equals three-fifths into one of our equations.
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Iβve chosen equation one.
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But it could be either of them.
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So when we substitute π equals three-fifths into equation one, we get four multiplied by three-fifths plus π is equal to negative one which is gonna give us twelve-fifths because we multiplied four by the numerator.
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So weβve got twelve-fifths plus π is equal to negative one.
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So now, if we subtract twelve-fifths from both sides of the equation, weβre gonna get π is equal to negative seventeen-fifths.
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And we got that as a value for π because we had negative one minus twelve-fifths.
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Well, we can think of one as five-fifths.
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So, therefore, weβd have negative five-fifths minus twelve-fifths which gives us negative seventeen-fifths.
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So therefore if we substitute our values of π and π, which are π is equal to three-fifths and π is equal to negative seventeen-fifths, into the linear function general form, then weβll get π of π₯, because we are using the function notation, is equal to negative seventeen-fifths plus three-fifths π₯.
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Or it can be written as π of π₯ is equal to three-fifths π₯ minus seventeen-fifths.