Answer
$(1, 1, 1)$
Work Step by Step
Label the original equations first:
1. $3x - y + z = 3$
2. $x + y + 2z = 4$
3. $x + 2y + z = 4$
Combine equations $1$ and $2$ to eliminate the $y$ variable:
1. $3x - y + z = 3$
2. $x + y + 2z = 4$
Add the equations:
4. $4x + 3z = 7$
Combine equations $2$ and $3$ to eliminate the $y$ variable. Modify equation $2$ by multiplying it by $-2$:
5. $-2(x + y + 2z) = -2(4)$
Use distributive property:
5. $-2x - 2y - 4z = -8$
Combine equations $3$ and $5$:
3. $x + 2y + z = 4$
5. $-2x - 2y - 4z = -8$
Add the equations together:
7. $-x - 3z = -4$
Combine equations $4$ and $7$:
4. $4x + 3z = 7$
7. $-x - 3z = -4$
Add the two equations:
$3x = 3$
Divide each side of the equation by $3$:
$x = 1$
Substitute this $x$ value into equation $7$ to find the value for $z$:
$-1 - 3z = -4$
Add $1$ to both sides of the equation:
$-3z = -3$
Divide both sides by $-3$:
$z = 1$
Substitute the values for $x$ and $z$ into one of the original equations to find $y$. Let's use equation $3$:
3. $1 + 2y + 1 = 4$
Combine like terms on the left side of the equation:
$2y + 2 = 4$
Subtract $2$ from each side of the equation:
$2y = 2$
Divide both sides by $2$:
$y = 1$
The solution is $(1, 1, 1)$.
To check the solution, plug in the three values into one of the original equations. Use equation $2$:
$1 + 1 + 2(1) = 4$
$1 + 1 + 2 = 4$
$4 = 4$
The sides are equal to one another; therefore, the solution is correct.
