Question

show that the equations 3x+4y+5z=a,4x+5y+6z=b,5x+6y+7z=c do not have a solution.unless a+c=2b​

Answer 1

Answer:

The solution of the given system of equations is  $\left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}0\\4\\-3\end{array}\right]$

Step-by-step explanation:

Given: The system of equations

3x + 4y + 5z = a

4x + 5y + 6z = b

5x + 6y + 7z = c and ( a + c = 2b)

To Find: Solution of the given system of equations for (a,b,c) = (1,2,3)

Finding the solution of the given system of equations

For the given equation, we can write them in the form of a matrix such that AX = B;

$\left[\begin{array}{ccc}3&5&5\\4&5&6\\5&6&7\end{array}\right] = \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}a\\b\\c\end{array}\right]$

Since we are given that the solution of possible if a+c=2b, and for the given values (a,b,c) = (1,2,3), the following condition is satisfied. Therefore, solution of the system of equations at (a,b,c) = (1,2,3) can be found as,

$\left[\begin{array}{ccc}3&5&5\\4&5&6\\5&6&7\end{array}\right] = \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}1\\2\\3\end{array}\right]$

z = -3

y + 2z = -2

       y = 4

x + y + z = 1

       x = 0

Hence, the solution of the system of the given equations is,

$\left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}0\\4\\-3\end{array}\right]$

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