# Question Video: Applying Operations on Matrices to Find the Values of Unknowns Mathematics

Given that [π₯ β 5, β9 and β3π§, 1] + 5[5π₯, β6 and βπ§, 7] = [β3π₯ + 2, 4π¦ and β2, 2π] + 4[π₯, 3π¦ and β2, 4π], find the values of π₯, π¦, π§, and π.

03:22

### Video Transcript

Given that the matrix π₯ minus five, negative nine, negative three π§, one plus five multiplied by the matrix five π₯, negative six, negative π§, seven equals the matrix negative three π₯ plus two, four π¦, negative two, two π plus four multiplied by the matrix π₯, three π¦, negative two, four π, find the values of π₯, π¦, π§, and π.

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 weβve done that, what weβre gonna have is a matrix π₯ minus five, negative nine, negative three π§, one plus the matrix 25π₯, negative 30, negative five π§, 35 equals the matrix negative three π₯ plus two, four π¦, negative two, two π plus the matrix four π₯, 12π¦, negative eight, 16π.

So now what weβre gonna do to work out what our different values are, π₯, π¦, π§, and π, is in fact equate the corresponding elements. So first of all, weβre gonna look at our first element in each of our matrices. And what we can say is that π₯ minus five plus 25π₯ is gonna be equal to negative three π₯ plus two plus four π₯. So, weβre gonna have 26π₯ minus five equals π₯ plus two. So, now, if we subtract π₯ and add five to each side of the equation, weβre gonna get 25π₯ equals seven. And then if we divide through by 25, we get π₯ equals seven over 25.

Okay, great. So, we found our π₯-value. So, now, what we can do is move on to our next element to find our π¦-value. So, what we have is negative nine minus 30 is equal to four π¦ plus 12π¦. So, we get negative 39 equals 16π¦ and then divide through by 16. And when we do that, we get negative 39 over 16 equals π¦. Okay, so great. Weβve found our π¦-value.

So, now, what we can do is we can move on to our π§-value. And when we do that, what weβre gonna get is negative three π§ minus five π§ equals negative two minus eight. So, we get negative eight π§ equals negative 10. And then, we can divide through by negative eight, and we get π§ is equal to 10 over eight. And then, we can divide the numerator and denominator by two. And when we do that, we get π§ 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 π. And when we do that, what we get is one plus 35 equals two π plus 16π. So, we get 36 equals 18π. And then we can divide through by 18, and we get two is equal to π.

So, therefore, we can say that the solutions to the question are π₯ equals seven over 25, π¦ equals negative 39 over 16, π§ equals five over four, and π equals two.