# 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, โ1 โ1 โ1, 2 5 6, multiplied by ๐, ๐, ๐, = 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, ๐, 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-coefficients. And our third column represents our ๐-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. And again, this is because we look in our ๐ column and we see that the coefficient is two. And then finally, weโre gonna have plus four ๐. 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. And again, this is because Iโve got negative one in our middle ๐ column and then finally minus ๐. 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 plus six ๐ 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 plus four ๐ equals four; our second one, which is negative ๐ minus ๐ minus ๐ is equal to 14; and finally two ๐ plus five ๐ plus six ๐ equals 10.