Solve the system of equations below using either elimination, substitution or Crammer's rule: 3x+2y−5x= −10…

Question

Solve the system of equations below using either elimination, substitution or Crammer's rule: 3x+2y−5x= −10, 6x-z=8, −y+3z=−2

ScanSolve AI Tutor · Accepted Answer

Here's the solution to Question Five.

We are given the following system of equations:

Express $z$ in terms of $x$ from equation (2).
From equation (2):
$6x - z = 8$ $z = 6x - 8 \quad \text{(Equation 4)}$
Express $y$ in terms of $z$ from equation (3).
From equation (3):
$-y + 3z = -2$ $y = 3z + 2 \quad \text{(Equation 5)}$
Substitute Equation (4) into Equation (5) to express $y$ in terms of $x$ .
Substitute $z = 6x - 8$ into Equation (5):
$y = 3(6x - 8) + 2$ $y = 18x - 24 + 2$ $y = 18x - 22 \quad \text{(Equation 6)}$
Substitute Equation (4) and Equation (6) into Equation (1).
Substitute $z = 6x - 8$ and $y = 18x - 22$ into Equation (1):
$3x + 2(18x - 22) - 5(6x - 8) = -10$ $3x + 36x - 44 - 30x + 40 = -10$
Combine like terms:
$(3x + 36x - 30x) + (-44 + 40) = -10$ $9x - 4 = -10$
Solve for $x$ .
$9x = -10 + 4$ $9x = -6$ $x = -\frac{6}{9}$ $x = -\frac{2}{3}$
Substitute the value of $x$ back into Equation (4) to find $z$ .
$z = 6x - 8$ $z = 6\left(-\frac{2}{3}\right) - 8$ $z = -4 - 8$ $z = -12$
Substitute the value of $z$ back into Equation (5) to find $y$ .
$y = 3z + 2$ $y = 3(-12) + 2$ $y = -36 + 2$ $y = -34$
The solution to the system of equations is:
$x = -\frac{2}{3}, y = -34, z = -12$

Solve the system of equations below using either elimination, substitution or Crammer's rule: 3x+2y-5x= -10, 6x−z=8, -y+3z=-2