Question

solve the system.\\(\

$$\begin{cases}3x + 2y - 5z = -57\\\\-4x + 15y + 3z = -38\\\\4x + 3y + z = -24\\end{cases}$$

\\)enter your answer as an ordered triple \\((x, y, z)\\).\\((?,\square,\square)\\)

Explanation:

Step1: Eliminate \(x\) from equations 2 and 3

Add the second equation \(-4x + 15y + 3z = -38\) and the third equation \(4x + 3y + z = -24\):
\((-4x + 4x)+(15y + 3y)+(3z + z)=-38 + (-24)\)
\(18y + 4z = -62\) (Equation 4)

Step2: Eliminate \(x\) from equations 1 and 3

Multiply the third equation \(4x + 3y + z = -24\) by 3: \(12x + 9y + 3z = -72\) (Equation 5)
Multiply the first equation \(3x + 2y - 5z = -57\) by 4: \(12x + 8y - 20z = -228\) (Equation 6)
Subtract Equation 6 from Equation 5:
\((12x - 12x)+(9y - 8y)+(3z - (-20z))=-72 - (-228)\)
\(y + 23z = 156\) (Equation 7)

Step3: Solve Equation 7 for \(y\)

\(y = 156 - 23z\)

Step4: Substitute \(y = 156 - 23z\) into Equation 4

\(18(156 - 23z)+4z = -62\)
\(2808 - 414z + 4z = -62\)
\(2808 - 410z = -62\)
\(-410z = -62 - 2808\)
\(-410z = -2870\)
\(z = \frac{-2870}{-410}=7\)

Step5: Substitute \(z = 7\) into Equation 7 to find \(y\)

\(y = 156 - 23\times7\)
\(y = 156 - 161=-5\)

Step6: Substitute \(y = -5\) and \(z = 7\) into the third equation to find \(x\)

\(4x + 3\times(-5)+7 = -24\)
\(4x - 15 + 7 = -24\)
\(4x - 8 = -24\)
\(4x = -24 + 8\)
\(4x = -16\)
\(x = -4\)

Answer:

\((-4, -5, 7)\)