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, โˆ’1 โˆ’1 โˆ’1, 2 5 6, multiplied by ๐‘, ๐‘ž, ๐‘Ÿ, = 4, 14, 10.

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Video Transcript

Write down the set of simultaneous equations that could be solved using the matrix equation. Iโ€™ve got two, negative one, two, two, negative one, five, four, negative one, six multiplied by ๐‘, ๐‘ž, ๐‘Ÿ, is equal to four, 14, and 10.

When trying to work out which simultaneous equations could be solved, weโ€™re gonna start by looking at our first matrix and looking at each column. Our first column represents the ๐‘-coefficients. Our second column represents our ๐‘ž-coefficients. And our third column represents our ๐‘Ÿ-coefficients. So this is gonna be really useful now when weโ€™re actually looking to form our equations. And also if we look at our first matrix, we can say that itโ€™s a three-by-three matrix. And this helps tell us that thereโ€™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 weโ€™ve now got all the information we need. So letโ€™s form our simultaneous equations.

Our first term in our first equation is going to be two ๐‘. And thatโ€™s because weโ€™ve got a coefficient of two here. And then thatโ€™s two ๐‘ because as we said, the first column is all our coefficients of ๐‘. But also if youโ€™re looking at whatโ€™s happening, we actually multiply the term by ๐‘ in the second matrix.

Our second term in our first equation is gonna be positive two ๐‘ž. And again, this is because we look in our ๐‘ž column and we see that the coefficient is two. And then finally, weโ€™re gonna have plus four ๐‘Ÿ. And then from the answer matrix, we can say that this is all gonna be equal to four. Okay, fantastic! Weโ€™ve got our first simultaneous equation.

For our second equation, weโ€™re gonna start off with negative ๐‘. And be careful again with positives and negatives. And this is if weโ€™re looking, our first matrix, we can see that itโ€™s a negative one. So thatโ€™s why weโ€™ve got negative ๐‘. And then we have minus ๐‘ž. And again, this is because Iโ€™ve got negative one in our middle ๐‘ž column and then finally minus ๐‘Ÿ. And this is all gonna be equal to 14, which we again get from our answer matrix.

So, great! Weโ€™ve now got two simultaneous equations. Now move on to the final one. Weโ€™re gonna have two ๐‘ plus five ๐‘ž plus six ๐‘Ÿ 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 ๐‘ plus two ๐‘ž plus four ๐‘Ÿ equals four; our second one, which is negative ๐‘ minus ๐‘ž minus ๐‘Ÿ is equal to 14; and finally two ๐‘ plus five ๐‘ž plus six ๐‘Ÿ equals 10.

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