Question Video: Solving an Equation with One Root by Graphing Mathematics

The graph in the diagram represents the function 𝑓(𝑥) = −5𝑥 + (3/2). Without performing any calculation, use the graph to solve −5𝑥 + (3/2) = 0.

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Video Transcript

The graph in the diagram represents the function 𝑓 of 𝑥 equals negative five 𝑥 plus three over two. Without performing any calculation, use the graph to solve negative five 𝑥 plus three over two equals zero.

The key to this question is that we’re told to solve the equation without performing any calculation. So therefore, all we’re gonna use is the graph that we’ve got. Well, if we’ve got the equation negative five 𝑥 plus three over two equals zero, and we know that our line represents 𝑓 of 𝑥 is equal to negative five 𝑥 plus three over two. Well, then, what we’re looking for is the point where 𝑦 is equal to zero. And that’s because 𝑓 of 𝑥 is represented by 𝑦; they’re interchangeable.

So therefore, if we think about our equation being 𝑦 is equal to negative five 𝑥 plus three over two, then if we’re looking for the solution to negative five 𝑥 plus three over two is equal to zero. Then, 𝑦 is going to be equal zero. And we can see that on our graph, this is the point on the line where 𝑦 is equal to zero. Well, then, what we can do is read off our 𝑥-coordinate cause this is gonna be our 𝑥-value, or our solution. But before we do that, we need to know what the scale of our graph is.

And if we take a look between two of our points — so, for instance, here we’ve got one and 1.5 — we can see that there are five spaces. So therefore, 0.5 is worth five spaces. So, if we want to find out what one space is, we divide 0.5 by five, which gives us 0.1. Well, we can see that our point on the line is three spaces away from zero. So therefore, we can see that 𝑥 is gonna be equal to 0.3. And we’ve solved that without doing any calculation. All we’ve used is our graph.

Okay, great, we’ve solved it. But what we can do now is just plug this value back into our equation to check it works. So, this is just a way of checking. So, we’ve already done it without calculation. But we’re now going to use a calculation just to check our answer.

So, to check, what we do is we substitute in 𝑥 equals 0.3. So, we’re gonna do that. We’re gonna get negative five multiplied by 0.3 plus three over two is equal to zero. So, when we do that, we get negative 1.5 plus three over two is equal to zero. So, now, what we would do is we convert either 1.5 into a fraction or three over two into a decimal. Well, I’ve converted three over two into a decimal cause I know it’s 1.5. So, then, if we have negative 1.5 plus 1.5, this is gonna be equal to zero cause they’re gonna cancel each other out. So therefore, we can say yes, the correct answer is 𝑥 is equal to 0.3.

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