# Video: Identifying a Set of Simultaneous Equations from a Matrix Equation

Write down the set of simultaneous equations that could be solved using the matrix equation 2 2 4, −1 −1 −1, 2 5 6, multiplied by 𝑝, 𝑞, 𝑟, = 4, 14, 10.

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### Video Transcript

Write down the set of simultaneous equations that could be solved using the matrix equation. I’ve got two, negative one, two, two, negative one, five, four, negative one, six multiplied by 𝑝, 𝑞, 𝑟, is equal to four, 14, and 10.

When trying to work out which simultaneous equations could be solved, we’re gonna start by looking at our first matrix and looking at each column. Our first column represents the 𝑝-coefficients. Our second column represents our 𝑞-coefficients. And our third column represents our 𝑟-coefficients. So this is gonna be really useful now when we’re actually looking to form our equations. And also if we look at our first matrix, we can say that it’s a three-by-three matrix. And this helps tell us that there’s gonna be three simultaneous equations that we need to find.

Now if we take a look at the answer matrix, what this tells us, this actually tells us what values our simultaneous equations are gonna be equal to. Okay, great! So we’ve now got all the information we need. So let’s form our simultaneous equations.

Our first term in our first equation is going to be two 𝑝. And that’s because we’ve got a coefficient of two here. And then that’s two 𝑝 because as we said, the first column is all our coefficients of 𝑝. But also if you’re looking at what’s happening, we actually multiply the term by 𝑝 in the second matrix.

Our second term in our first equation is gonna be positive two 𝑞. And again, this is because we look in our 𝑞 column and we see that the coefficient is two. And then finally, we’re gonna have plus four 𝑟. And then from the answer matrix, we can say that this is all gonna be equal to four. Okay, fantastic! We’ve got our first simultaneous equation.

For our second equation, we’re gonna start off with negative 𝑝. And be careful again with positives and negatives. And this is if we’re looking, our first matrix, we can see that it’s a negative one. So that’s why we’ve got negative 𝑝. And then we have minus 𝑞. And again, this is because I’ve got negative one in our middle 𝑞 column and then finally minus 𝑟. And this is all gonna be equal to 14, which we again get from our answer matrix.

So, great! We’ve now got two simultaneous equations. Now move on to the final one. We’re gonna have two 𝑝 plus five 𝑞 plus six 𝑟 is equal to 10. Okay, great! So we can now say that the set of simultaneous equations that could be solved by a matrix equation are one, which is two 𝑝 plus two 𝑞 plus four 𝑟 equals four; our second one, which is negative 𝑝 minus 𝑞 minus 𝑟 is equal to 14; and finally two 𝑝 plus five 𝑞 plus six 𝑟 equals 10.