Solve the system of equations by using the inverse of the coefficient matrix of the equivalent matrix equation.
1 answer:
The solution for the given system of equations x + 8y = -37, 4x + 8y = -52 is ![\left[\begin{array}{ccc}&5&\\\\&4&\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%265%26%5C%5C%5C%5C%264%26%5Cend%7Barray%7D%5Cright%5D)
Given,
System of equations as,
x + 8y = -37
4x + 8y = -52
We have to solve this by using the inverse of coefficient matrix of the equivalent matrix equation.
That is,
![A=\left[\begin{array}{ccc}a&&b\\\\c&&d\end{array}\right]](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Da%26%26b%5C%5C%5C%5Cc%26%26d%5Cend%7Barray%7D%5Cright%5D)
![A^{-1} =\frac{1}{ad -bc} \left[\begin{array}{ccc}d&&-b\\\\-c&&a\end{array}\right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%20%3D%5Cfrac%7B1%7D%7Bad%20-bc%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dd%26%26-b%5C%5C%5C%5C-c%26%26a%5Cend%7Barray%7D%5Cright%5D)
Now we can solve the equations.
Here we have,
x + 8y = -37
4x + 8y = -52
Now in matrix form,
![=\left[\begin{array}{ccc}&-37&\\\\&-52&\end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%26-37%26%5C%5C%5C%5C%26-52%26%5Cend%7Barray%7D%5Cright%5D)
A X B
We know that,
![A^{-1} =\frac{1}{ad -bc} \left[\begin{array}{ccc}d&&-b\\\\-c&&a\end{array}\right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%20%3D%5Cfrac%7B1%7D%7Bad%20-bc%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dd%26%26-b%5C%5C%5C%5C-c%26%26a%5Cend%7Barray%7D%5Cright%5D)
Therefore,
![A^{-1} = \frac{1}{(1X8)-(4X8)} \left[\begin{array}{ccc}8&&-8\\\\-4&&1\end{array}\right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%20%3D%20%5Cfrac%7B1%7D%7B%281X8%29-%284X8%29%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D8%26%26-8%5C%5C%5C%5C-4%26%261%5Cend%7Barray%7D%5Cright%5D)
![=\frac{1}{8-32} \left[\begin{array}{ccc}8&&-8\\\\-4&&1\end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B8-32%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D8%26%26-8%5C%5C%5C%5C-4%26%261%5Cend%7Barray%7D%5Cright%5D)
![=\frac{1}{-24} \left[\begin{array}{ccc}8&&-8\\\\-4&&1\end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B-24%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D8%26%26-8%5C%5C%5C%5C-4%26%261%5Cend%7Barray%7D%5Cright%5D)
![=\left[\begin{array}{ccc}\frac{-8}{24} &&\frac{8}{24} \\\\\frac{4}{24} &&\frac{-1}{24} \end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Cfrac%7B-8%7D%7B24%7D%20%26%26%5Cfrac%7B8%7D%7B24%7D%20%5C%5C%5C%5C%5Cfrac%7B4%7D%7B24%7D%20%26%26%5Cfrac%7B-1%7D%7B24%7D%20%5Cend%7Barray%7D%5Cright%5D)
Then,
![\left[\begin{array}{ccc}&x&\\\\&y&\end{array}\right] =](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%26x%26%5C%5C%5C%5C%26y%26%5Cend%7Barray%7D%5Cright%5D%20%3D)
![\left[\begin{array}{ccc}&\frac{37}{52} &\\\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%26%5Cfrac%7B37%7D%7B52%7D%20%26%5C%5C%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![=\frac{1}{24} \left[\begin{array}{ccc}-8&&8\\\\4&&-1\end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B24%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-8%26%268%5C%5C%5C%5C4%26%26-1%5Cend%7Barray%7D%5Cright%5D)
![\left[\begin{array}{ccc}&\frac{37}{52} &\\\\\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%26%5Cfrac%7B37%7D%7B52%7D%20%26%5C%5C%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![=\frac{1}{24} \left[\begin{array}{ccc}(-8X37)+(8X52)\\\\(4X37)+(-1X52)\end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B24%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%28-8X37%29%2B%288X52%29%5C%5C%5C%5C%284X37%29%2B%28-1X52%29%5Cend%7Barray%7D%5Cright%5D)
![=\frac{1}{24} \left[\begin{array}{ccc}-296+416\\\\148-52\end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B24%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-296%2B416%5C%5C%5C%5C148-52%5Cend%7Barray%7D%5Cright%5D)
![=\frac{1}{24} \left[\begin{array}{ccc}120\\\\96\end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B24%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D120%5C%5C%5C%5C96%5Cend%7Barray%7D%5Cright%5D)
![=\left[\begin{array}{ccc}\frac{120}{24} \\\\\frac{96}{24} \end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Cfrac%7B120%7D%7B24%7D%20%5C%5C%5C%5C%5Cfrac%7B96%7D%7B24%7D%20%5Cend%7Barray%7D%5Cright%5D)
![=\left[\begin{array}{ccc}5\\\\4\end{array}\right]](https://tex.z-dn.net/?f=%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%5C%5C%5C%5C4%5Cend%7Barray%7D%5Cright%5D)
That is ![\left[\begin{array}{ccc}x\\\\y\end{array}\right] =\left[\begin{array}{ccc}5\\\\4\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5C%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%5C%5C%5C%5C4%5Cend%7Barray%7D%5Cright%5D)
Learn more about matrix equations here: brainly.com/question/27799804
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The question is incomplete. Completed question is given below:
Solve The System Of Equations By Using The Inverse Of The Coefficient Matrix Of The Equivalent Matrix Equation.
x + 8y = -37
4x + 8y = -52
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