Video: Solving Problems Leading to Linear Equations

Solve 5(3^(1 − 𝑥)) = 45 for 𝑥.

03:10

Video Transcript

Solve five multiplied by three to the power of one minus 𝑥 equals 45 for 𝑥.

So if we want to solve the equation five multiplied by three to the power of one minus 𝑥 equals 45, then the first thing we want to do is divide each side of the equation by five. And when we do that, what we’re left with is three to the power of one minus 𝑥 equals nine. And that’s because if we have five multiplied by something and then divide by five, we’re just left with that something, which in this case is three to the power of one minus 𝑥. And if we divide 45 by five, we get nine.

So now, what we do next because we’ve got our 𝑥 as part of the exponent or power of three? So what’s the next part of the process? Well, what we can look to to do, and we can do it with this example, is have the base number being the same on the left- and right-hand side of our equation. So, for instance, on the left-hand side of the equation, we’ve got three as our base number cause it’s three to the power of one minus 𝑥. So therefore, we quite like to have a three as our base on the right-hand side of the equation. And we can, because we can rewrite nine as three multiplied by three or three squared. So therefore, what we can do is rewrite our equation as three to the power of one minus 𝑥 equals three squared or three raised to the power of two.

So now, we’ve got this and we’ve got the same base. What we can do is we can equate the exponents. And when we do that, what we have is one minus 𝑥 is equal to two. And so that we have a positive 𝑥, the next step could be to add 𝑥 to each side of our equation. And this gives us one equals 𝑥 plus two. And we get that cause if we have one minus 𝑥 add 𝑥, we get one. And therefore, if we got two add 𝑥, we can get 𝑥 plus two. And then, the next step would be to subtract two from both sides of the equation. So we want to find out what 𝑥 is. And when we do that, we get negative one is equal 𝑥. So the solution to five multiplied by three to the power of one minus 𝑥 equals 45 is 𝑥 equals negative one.

Now, it’s worth noting that, at this stage in our equation, we could’ve subtracted one from each side of the equation rather adding 𝑥. And this would’ve given us negative 𝑥 is equal to one. Now, I tend to do the other way just so that we don’t have a negative 𝑥 because it can sometimes confuse. But all we have to do now if we want to find 𝑥 from this is divide through by negative one. And if we do this, we end up with the same answer, which is 𝑥 is equal to negative one. Okay, great. So we’ve solved the problem.

What we can do is we can double check our answer by substituting back in the value we found for 𝑥. So if we do that, what we’d have is five multiplied by three to the power of one minus negative one is equal to 45. Well, if we minus a negative, it’s the same as add. So we get five multiplied by three squared equals 45. Well, three squared is nine. So five multiplied by nine is 45 is correct. So we’ve double checked our answer. And it definitely is 𝑥 equals negative one.

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