Solve the system of linear equations using the elimination method.
-$$\begin{aligned}
7 x - 5 y - z & = 27 \
x - 8 y + 9 z & = 4 \
3 x + y + z & = 33
\end{aligned}$$
A)(8, 4, 5)
B)(-8, 5, 16)
C)(16, 5, -8)
D)(8, 5, 4)

Question

Quizplus · Accepted Answer

To solve the system using the elimination method, we can start by eliminating one variable from two pairs of equations, then solve the resulting system of two equations with two variables.First, let's eliminate $z$ by combining the first and second equations, and the first and third equations.From the first and second equations:$$7x - 5y - z = 27$$$$x - 8y + 9z = 4$$Multiplying the first equation by 9 to match the $z$ coefficient in the second equation:$$63x - 45y - 9z = 243$$$$x - 8y + 9z = 4$$Adding these equations to eliminate $z$:$$64x - 53y = 247$$ (Equation 4)From the first and third equations:$$7x - 5y - z = 27$$$$3x + y + z = 33$$Adding these equations directly to eliminate $z$:$$10x - 4y = 60$$ (Equation 5)Now, we solve the system formed by Equations 4 and 5. To simplify the process, let's multiply Equation 5 by 13.25 (to match the $x$ coefficient in Equation 4 when simplified):$$132.5x - 53y = 795$$Now, we can subtract Equation 4 from this new equation to solve for $x$:$$132.5x - 53y = 795$$$$-(64x - 53y = 247)$$$$68.5x = 548$$$$x = 8$$Substituting $x = 8$ into Equation 5:$$10(8) - 4y = 60$$$$80 - 4y = 60$$$$-4y = -20$$$$y = 5$$Finally, substitute $x = 8$ and $y = 5$ into one of the original equations to solve for $z$. Using the first equation:$$7(8) - 5(5) - z = 27$$$$56 - 25 - z = 27$$$$31 - z = 27$$$$-z = -4$$$$z = 4$$Therefore, the solution is $x = 8$, $y = 5$, $z = 4$, which corresponds to choice D.

Solve the System of Linear Equations Using the Elimination Method 7x−5y−z=27x−8y+9z=43x+y+z=33\begin{aligned}7 x - 5 y - z & = 27 \\x - 8 y + 9 z & = 4 \\3 x + y + z & = 33\end{aligned}7x−5y−zx−8y+9z3x+y+z​=27=4=33​

Solve the System of Linear Equations Using the Elimination Method $\begin{aligned}7 x - 5 y - z & = 27 \\x - 8 y + 9 z & = 4 \\3 x + y + z & = 33\end{aligned}$