Question

solve the system.\

$$\begin{cases}4x - 5y + z = -5\\\\-3x + 5y + 3z = -10\\\\5x + 3y - 8z = 114\\end{cases}$$

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

Explanation:

Step1: Eliminate \( y \) from first two equations

Add the first equation \( 4x - 5y + z = -5 \) and the second equation \( -3x + 5y + 3z = -10 \):
\( (4x - 3x)+(-5y + 5y)+(z + 3z)=-5 + (-10) \)
\( x + 4z=-15 \) --- Equation (4)

Step2: Manipulate equations to eliminate \( y \) again

Multiply the first equation \( 4x - 5y + z = -5 \) by 3: \( 12x - 15y + 3z = -15 \) --- Equation (5)
Multiply the second equation \( -3x + 5y + 3z = -10 \) by 3: \( -9x + 15y + 9z = -30 \) --- Equation (6)
Wait, better to use first and third. Multiply first equation by 3: \( 12x - 15y + 3z = -15 \) (Equation 5)
Third equation: \( 5x + 3y - 8z = 114 \), multiply by 5: \( 25x + 15y - 40z = 570 \) --- Equation (7)
Add Equation (5) and Equation (7):
\( (12x + 25x)+(-15y + 15y)+(3z - 40z)=-15 + 570 \)
\( 37x - 37z = 555 \)
Divide by 37: \( x - z = 15 \) --- Equation (8)

Step3: Solve for \( x \) and \( z \)

We have Equation (4): \( x + 4z=-15 \)
Equation (8): \( x - z = 15 \)
Subtract Equation (8) from Equation (4):
\( (x + 4z)-(x - z)=-15 - 15 \)
\( x + 4z - x + z=-30 \)
\( 5z=-30 \)
\( z = -6 \)

Substitute \( z = -6 \) into Equation (8): \( x - (-6)=15 \)
\( x + 6 = 15 \)
\( x = 9 \)

Step4: Solve for \( y \)

Substitute \( x = 9 \), \( z = -6 \) into first equation \( 4x - 5y + z = -5 \):
\( 4(9)-5y + (-6)=-5 \)
\( 36 - 5y - 6=-5 \)
\( 30 - 5y=-5 \)
\( -5y=-35 \)
\( y = 7 \)

Answer:

\((9, 7, -6)\)