Solve the system. {:[x+5y-z=,-6],[-4x-y+z=,18],[x-y+5z=,6]:}

Question

Solve the system.  {:[x+5y-z=,-6],[-4x-y+z=,18],[x-y+5z=,6]:}

Bytelearn · Accepted Answer

Write Equations: Write down the system of equations to be solved. $$ \begin{align*} x + 5y - z &= -6 \ -4x - y + z &= 18 \ x - y + 5z &= 6 \end{align*} $$Eliminate Variable x: Choose two equations to eliminate one variable. We will use the first and the third equations to eliminate the variable $x$. $$ \begin{align*} x + 5y - z &= -6 \ x - y + 5z &= 6 \end{align*} $$ Subtract the third equation from the first to eliminate $x$. $$ (5y - z) - (-y + 5z) = -6 - 6 $$Combine Equations: Perform the subtraction to find an equation in terms of $y$ and $z$. $$ 5y - z + y - 5z = -12 $$ Combine like terms. $$ 6y - 6z = -12 $$Simplify Equation: Simplify the equation by dividing all terms by 6. $$ y - z = -2 $$ This is our fourth equation.Eliminate Variable x: Now, choose two other equations to eliminate the same variable $x$. We will use the second and the third equations. $$ \begin{align*} -4x - y + z &= 18 \ x - y + 5z &= 6 \end{align*} $$ Multiply the third equation by 4 to make the coefficients of $x$ opposites. $$ \begin{align*} -4x - y + z &= 18 \ 4x - 4y + 20z &= 24 \end{align*} $$Combine Equations: Add the modified third equation to the second to eliminate $x$. $$ (-4x - y + z) + (4x - 4y + 20z) = 18 + 24 $$ Perform the addition. $$ -5y + 21z = 42 $$Simplify Equation: Simplify the equation by dividing all terms by -5. $$ y - \frac{21}{5}z = -\frac{42}{5} $$ This is our fifth equation.Eliminate Variable y: We now have two equations with just $y$ and $z$: $$ \begin{align*} y - z &= -2 \ y - \frac{21}{5}z &= -\frac{42}{5} \end{align*} $$ Subtract the first equation from the second to eliminate $y$. $$ \left(y - \frac{21}{5}z\right) - (y - z) = -\frac{42}{5} - (-2) $$Combine Equations: Perform the subtraction to find an equation in terms of $z$. $$ -\frac{21}{5}z + z = -\frac{42}{5} + 2 $$ Combine like terms. $$ -\frac{16}{5}z = -\frac{32}{5} $$Solve for z: Solve for $z$ by dividing both sides by $-\frac{16}{5}$. $$ z = \frac{-\frac{32}{5}}{-\frac{16}{5}} = 2 $$Find y: Substitute $z = 2$ into the fourth equation to find $y$. $$ y - z = -2 $$ $$ y - 2 = -2 $$ Add 2 to both sides. $$ y = 0 $$Substitute and Find x: Substitute $y = 0$ and $z = 2$ into one of the original equations to find $x$. We will use the first equation. $$ x + 5y - z = -6 $$ $$ x + 5(0) - 2 = -6 $$ Simplify. $$ x - 2 = -6 $$ Add 2 to both sides. $$ x = -4 $$

Solve the system. $\newline$ $\begin{array}{rr} x+5 y-z= & -6 \\ -4 x-y+z= & 18 \\ x-y+5 z= & 6 \end{array}$

Full solution

More problems from Write an equation word problem

Related topics

Solve the system.\newlinex+5y−z=amp;−6−4x−y+z=amp;18x−y+5z=amp;6 \begin{array}{rr} x+5 y-z= &amp; -6 \\ -4 x-y+z= &amp; 18 \\ x-y+5 z= &amp; 6 \end{array} x+5y−z=−4x−y+z=x−y+5z=​amp;−6amp;18amp;6​

Solve the system. $\newline$ $\begin{array}{rr} x+5 y-z= & -6 \\ -4 x-y+z= & 18 \\ x-y+5 z= & 6 \end{array}$