Find the values of the variables for which each statement is t...

Question

Find the values of the variables for which each statement is true, if possible.

$[\begin{matrix} - 7 + z & 4 r & 8 s \\ 6 p & 2 & 5 \end{matrix}] + [\begin{matrix} - 9 & 8 r & 3 \\ 2 & 5 & 4 \end{matrix}] = [\begin{matrix} 2 & 36 & 27 \\ 20 & 7 & 12 a \end{matrix}]$

Accepted Answer

Hello, everyone. We are going to determine if possible the values for the variables that satisfy the given statement, we have an equation that is set up with a two by three matrix added to a two by three matrix which results in a two by three matrix. The top line of the first matrix is negative eight plus X 16 minus Y three Z plus 11. The second row is seven W 82 and that's added to the other matrix where the top line is five negative 10 3. The second row is three negative and this equals the two by three matrix where the top row is 2023 the second row is 17 76. We have four answer choices giving our values for X Y Z W and T which we will find in this problem recall when you're adding or subtracting matrices that we can put corresponding elements together. So for example, starting from the top left corner and going across the row, I can say negative eight plus XX plus the top left corner element from the second matrix which is five has to equal the top left corner from the final matrix which is two. So I'm gonna solve this to find out what X is right now negative eight plus five is negative three plus X equals two, adding three to both sides. I get that X equals five, one variable down four to go. The next one I need to solve is for why? So that is the second element of the top row. So 16 minus Y, the corresponding element in the next matrix is negative 10. So I'm going to write that as minus 10 and this needs to equal the corresponding element in the answer matrix which is zero, combining the lake terms that are on the same side. I get I'm sorry, six minus Y equals zero, subtract six from both sides negative Y equals negative six dividing by negative one. We get that Y equals a positive six. All right. So we found that variable. The next variable I see is Z. So I'm gonna solve that again. It's the third element in the top row. So it'd be three Z plus 11 plus the third element in the top row from the next matrix which is three. And that needs to equal the third element in the top row from the answer matrix which is 23 combining my like terms. I get three Z plus 14 plus subtracting 14 from both sides. Three Z equals nine divided by three, Z must equal three Z three. Next I'm gonna look for my next variable which looks like seven W and that is the first element in the second row of the first matrix. So that would be seven W plus the first element of the second row of the next matrix, which is three equals the first element of the second row of the answer matrix which is solving this by subtracting three from both sides, I get seven, W equals 14, dividing by seven, get the W equals two. All right, we only have one more, we need to find. And that's the value T T is the third element in the second row of the answer matrix. So we're gonna start with the third element of the second row of the first matrix which is two plus the third element of the second row of the second matrix which is seven. And that needs to equal the third element of the second row of the answer matrix which is six T, so two plus seven has to equal six, T, two plus seven is nine. So nine equals six T dividing both sides by six, I get that T equals 9/6 which I will be reducing. And that should be three halves. So T equals three halves. So now that we have all five variables, let's see where they match. So X equals five. So that's answer choices B through C Y equals six. So now we're down to C or D Z equals three. Still answer choices C or D, uh, W equals two. Still between C and D P equals three halves that does it, it's answer choice. D have a nice day.

Find the values of the variables for which each statement is true, if possible.
$[\begin{matrix} - 7 + z & 4 r & 8 s \\ 6 p & 2 & 5 \end{matrix}] + [\begin{matrix} - 9 & 8 r & 3 \\ 2 & 5 & 4 \end{matrix}] = [\begin{matrix} 2 & 36 & 27 \\ 20 & 7 & 12 a \end{matrix}]$

Key Concepts

Matrix Addition

Matrix Dimensions

Solving Matrix Equations

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