Given that the matrix 𝑥 minus five, negative nine, negative three 𝑧, one plus five multiplied by the matrix five 𝑥, negative six, negative 𝑧, seven equals the matrix negative three 𝑥 plus two, four 𝑦, negative two, two 𝑘 plus four multiplied by the matrix 𝑥, three 𝑦, negative two, four 𝑘, find the values of 𝑥, 𝑦, 𝑧, and 𝑘.
So, the first thing we want to do when we take a look at our matrices is multiply those with a scalar, by the scalar value. And to do that, all we do is multiply the scalar value by each of the elements inside of our matrix. And once we’ve done that, what we’re gonna have is a matrix 𝑥 minus five, negative nine, negative three 𝑧, one plus the matrix 25𝑥, negative 30, negative five 𝑧, 35 equals the matrix negative three 𝑥 plus two, four 𝑦, negative two, two 𝑘 plus the matrix four 𝑥, 12𝑦, negative eight, 16𝑘.
So now what we’re gonna do to work out what our different values are, 𝑥, 𝑦, 𝑧, and 𝑘, is in fact equate the corresponding elements. So first of all, we’re gonna look at our first element in each of our matrices. And what we can say is that 𝑥 minus five plus 25𝑥 is gonna be equal to negative three 𝑥 plus two plus four 𝑥. So, we’re gonna have 26𝑥 minus five equals 𝑥 plus two. So, now, if we subtract 𝑥 and add five to each side of the equation, we’re gonna get 25𝑥 equals seven. And then if we divide through by 25, we get 𝑥 equals seven over 25.
Okay, great. So, we found our 𝑥-value. So, now, what we can do is move on to our next element to find our 𝑦-value. So, what we have is negative nine minus 30 is equal to four 𝑦 plus 12𝑦. So, we get negative 39 equals 16𝑦 and then divide through by 16. And when we do that, we get negative 39 over 16 equals 𝑦. Okay, so great. We’ve found our 𝑦-value.
So, now, what we can do is we can move on to our 𝑧-value. And when we do that, what we’re gonna get is negative three 𝑧 minus five 𝑧 equals negative two minus eight. So, we get negative eight 𝑧 equals negative 10. And then, we can divide through by negative eight, and we get 𝑧 is equal to 10 over eight. And then, we can divide the numerator and denominator by two. And when we do that, we get 𝑧 is equal to five over four or five-quarters.
So, now finally, what we can do is move on to our final element so we can find 𝑘. And when we do that, what we get is one plus 35 equals two 𝑘 plus 16𝑘. So, we get 36 equals 18𝑘. And then we can divide through by 18, and we get two is equal to 𝑘.
So, therefore, we can say that the solutions to the question are 𝑥 equals seven over 25, 𝑦 equals negative 39 over 16, 𝑧 equals five over four, and 𝑘 equals two.