Answer
$(1, 3, 2)$
Work Step by Step
Label the original equations:
1. $5x - y + z = 4$
2. $x + 2y - z = 5$
3. $2x + 3y - 3z = 5$
Solve equation $1$ for $z$. This will be equation $4$:
4. $z = -5x + y + 4$
Substitute the expression for $z$ given by equation $4$ into equation $2$:
$x + 2y - (-5x + y + 4) = 5$
Combine like terms. This will be equation $5$:
5. $6x + y - 4 = 5$
Move constants to the right side of the equation by adding $4$ to both sides of the equation:
5. $6x + y = 9$
Substitute the expression for $x$ given by equation $4$ into equation $3$:
$2x + 3y - 3(-5x + y + 4) = 5$
Distribute and multiply to simplify. This will be equation $6$:
6. $2x + 3y + 15x - 3y - 12 = 5$
Combine like terms on the left side of the equation:
6. $17x - 12 = 5$
Move constants to the right side of the equation by subtracting $10$ from both sides of the equation:
6. $17x = 17$
Divide both sides of the equation by $17$ to solve for $x$:
$x = 1$
Substitute this value for $x$ into equation $5$:
$6(1) + y = 9$
Subtract $6$ from both sides of the equation to solve for $y$:
$y = 3$
Substitute the values for $x$ and $y$ into equation $2$ to find the value of $z$:
$1 + 2(3) - z = 5$
Multiply to simplify:
$1 + 6 - z = 5$
Combine like terms on the left side of the equation:
$7 - z = 5$
Subtract $7$ from both sides of the equation to move constants to the right side of the equation:
$-z = -2$
Divide both sides of the equation by $-1$ to solve for $z$:
$z = 2$
The solution is $(1, 3, 2)$.
To check the solution, plug in the three values into one of the original equations. Use equation $3$:
$2(1) + 3(3) - 3(2) = 5$
$2 + 9 - 6 = 5$
$5 = 5$
The sides are equal to one another; therefore, the solution is correct.
