Answer
$(4, 1, 6)$
Work Step by Step
Label the original equations first:
1. $x - 4y + z = 6$
2. $2x + 5y - z = 7$
3. $2x - y - z = 1$
Solve equation $1$ for $z$. This will become equation $4$:
4. $z = -x + 4y + 6$
Substitute the expression for $z$ into equation $2$:
$2x + 5y - (-x + 4y + 6) = 7$
Use distributive property to simplify:
$2x + 5y + x - 4y - 6 = 7$
Combine like terms on the left side of the equation:
$3x + y - 6 = 7$
Add $6$ to both sides of the equation. This will become equation $5$:
5. $3x + y = 13$
Substitute the expression for $z$ into equation $3$:
$2x - y - (-x + 4y + 6) = 1$
Distribute to simplify:
$2x - y + x - 4y - 6 = 1$
Combine like terms:
$3x - 5y - 6 = 1$
Add $6$ to each side of the equation. This will become equation $6$:
6. $3x - 5y = 7$
Set up a system of equations made up of equations $5$ and $6$:
$3x + y = 13$
$3x - 5y = 7$
Multiply equation $5$ by $-1$:
5. $-1(3x + y) = -1(13)$
Distribute and multiply:
5. $-3x - y = -13$
Set up the system of equations:
5. $-3x - y = -13$
6. $3x - 5y = 7$
Add the equations:
$-6y = -6$
Divide both sides of the equation by $-6$ to solve for $y$:
$y = 1$
Substitute this value for $y$ into equation $6$ to find the value of $x$:
$3x - 5(1) = 7$
Multiply to simplify:
$3x - 5 = 7$
Add $5$ to each side of the equation:
$3x = 12$
Divide each side of the equation by $3$ to solve for $x$:
$x = 4$
Substitute the values for $x$ and $y$ into equation $3$ to find the value of $z$:
$2(4) - 1 - z = 1$
Multiply to simplify:
$8 - 1 - z = 1$
Add like terms on the left side of the equation:
$7 - z = 1$
Subtract $7$ from each side of the equation:
$-z = -6$
Divide each side by $-1$ to solve for $z$:
$z = 6$
The solution is $(4, 1, 6)$.
To check the solution, plug in the three values into one of the original equations. Use equation $2$:
$2(4) + 5(1) - 6 = 7$
$8 + 5 - 6 = 7$
$6 = 6$
The sides are equal to one another; therefore, the solution is correct.
