Question Video: Finding Unknown Elements of a Matrix Using Equality of Matrices Mathematics

Given that the matrix 9^(π‘₯), 9π‘₯ + 3𝑦 and 2π‘₯ βˆ’ 6𝑦, 9^(𝑦) = the matrix 32, 𝑏 and π‘Ž, 2. Find the value of 𝑏/π‘Ž.

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Video Transcript

Given that the matrix nine to the power of π‘₯, nine π‘₯ plus three 𝑦, two π‘₯ minus six 𝑦, nine to the power of 𝑦 is equal to the matrix 32, 𝑏, π‘Ž, two. Find the value of 𝑏 over π‘Ž or 𝑏 divided by π‘Ž.

So now we take a look at the properties behind a matrix being equal to another matrix. First of all, they have to have the same number of columns and same number of rows, which we have. Then, each of the corresponding terms must be the same value. So, for example, nine to the power of π‘₯ must be equal to 32. And we can use this property to help us solve the problem. Because what we can do is we can set up a couple of equations to find π‘₯ and 𝑦.

I’m gonna start with π‘₯. And we can say that nine to the power of π‘₯ is equal to 32. So if we look and solve an equation like this, we need to think, well, how can we find π‘₯ because π‘₯ is the exponent. Well, we can solve it using logarithms. So we are gonna take the log of both sides of the equation. However, we’re not gonna take log to the base 10, which is regularly used. I’m gonna take log to the base nine. But why am I doing this? Well, I’m doing this because of one of our log laws. And that tells us that if we have log to the base π‘Ž of π‘Ž, it’s gonna be equal to one. So therefore, we’re gonna use log to the base nine because it gonna help us. Because we’re gonna have log to the base nine of nine at some point in our equation.

So now we have log to the base nine of nine to the power of π‘₯ equals log to the base nine of 32. So what could we do now? And now, we’re gonna use one of the other log rules. And this one is often called the parallel. And it tells us that if we have the log of π‘₯ to the power of 𝑦, then it’s gonna be equal to 𝑦 multiplied by the log of π‘₯. So when we use this, we get π‘₯ log to the base nine of nine equals log to the base nine of 32. And we already know that log to the base nine of nine is gonna be equal to one because we mentioned that earlier. So then, we’re left with π‘₯ is equal to log to the base nine of 32. Okay, great! So we found π‘₯. Now let’s find 𝑦.

Well, this time, we have nine to the of power 𝑦 equals two. So again, we’re gonna use the same method. So we’re gonna take log to the base nine of each side. So we get log to the base nine of nine to the power of 𝑦 equals log to the base nine of two. So then, we’re gonna use our log power rule. And so we’re gonna get 𝑦 log to the base nine of nine equals log to the base nine of two. So therefore, we’re left with 𝑦 is equal to log to the base nine of two. Okay, great! So we’ve now our π‘₯ and 𝑦. Let’s move on to the next stage of the question.

Well, when we move on to the next stage of the question, what we’re gonna be doing is combining π‘₯s and 𝑦s. So therefore, it’ll be useful if they’re in the same form. So let’s take a look at our π‘₯ and 𝑦. We have π‘₯ is equal to log to the base nine of 32. And 𝑦 is equal to log to the base nine of two. Well, if we look at 32 in terms of two, we can say that 32 is equal to two to the power of five. So therefore, we could rewrite π‘₯ as π‘₯ is equal to log to the base nine of two to the power five. So therefore, we can apply our power rule again. And we get π‘₯ is equal to five log to the base nine of two. Okay, great. So then now, in a very similar form, and this gonna be very useful in the next stage.

So now what we do is we move our attentions across to 𝑏 and π‘Ž. Because in the question, we want to find the value of 𝑏 divided by π‘Ž. But first of all, we need to know what 𝑏 and π‘Ž are. Well, we can see from our matrices that π‘Ž is gonna be equal to two π‘₯ minus six 𝑦. And that’s because they’re the corresponding terms in our matrices. So therefore, we can say that π‘Ž is equal to two multiplied by five log to the base nine of two, because that was our π‘₯, minus six multiplied by log to the base nine of two cause that’s our 𝑦. So that gives us 10 log to the base nine of two minus six log to the base nine of two. So therefore, π‘Ž is equal to four log to the base nine of two. Okay, great. So that’s π‘Ž. So now, let’s find 𝑏.

Well, 𝑏 is equal to nine π‘₯ plus three 𝑦. And that’s because they’re the corresponding terms in the matrices. So therefore, 𝑏 is gonna be equal to nine multiplied by five log to the base nine of two, and again that’s our π‘₯, plus three multiplied by log to the base nine of two because that’s our 𝑦. So 𝑏 is equal to 45 log to the base nine of two plus three log to the base nine of two. So therefore, we get a final result for 𝑏 of 48 log nine to the base of two. So great! We’ve found π‘₯ and 𝑦, then used these to find π‘Ž and 𝑏. So now let’s solve the problem.

So now to solve the problem, we need to do 𝑏 divided by π‘Ž. So we have 𝑏 divided by π‘Ž. This gonna be equal to 48 log to the base nine of two divided by four log to the base nine of two. Well, first of all, we can divide the numerator and denominator by log to the base nine of two. So these cancel out. So this gives us 48 divided by four. So therefore, we can say that given that the matrix nine to power of π‘₯, nine π‘₯ plus three 𝑦, two π‘₯ minus six 𝑦, nine to the power of 𝑦 is equal to the matrix 32, 𝑏, π‘Ž, two. Then, the value of 𝑏 divided by π‘Ž is equal to 12.

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