Solve for $x, y$ , and $z$ $\newline$ $\begin{array}{l} x+6 y-z=-5 \\ -2 x-5 y+2 z=3 \\ x-\frac{4}{5} y+z=\frac{19}{5} \end{array}$

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Geometry

Sine and cosine of complementary angles

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Q. Solve for

x, y

, and

z

\newline

\begin{array}{l} x+6 y-z=-5 \\ -2 x-5 y+2 z=3 \\ x-\frac{4}{5} y+z=\frac{19}{5} \end{array}

Write Equations: Write down the system of equations. $\newline$ We have the following system of linear equations: $\newline$ $1$ ) $x + 6y - z = -5$ $\newline$ $2$ ) $-2x - 5y + 2z = 3$ $\newline$ $3$ ) $x - \left(\frac{4}{5}\right)y + z = \left(\frac{19}{5}\right)$
Eliminate Fraction: Multiply the third equation by $5$ to eliminate the fraction. $\newline$ $5(x - (\frac{4}{5})y + z) = 5 \times (\frac{19}{5})$ $\newline$ This simplifies to: $\newline$ $5x - 4y + 5z = 19$ $\newline$ Now we have the system: $\newline$ $1$ ) $x + 6y - z = -5$ $\newline$ $2$ ) $-2x - 5y + 2z = 3$ $\newline$ $3$ ) $5x - 4y + 5z = 19$
Eliminate Variable $x$ : Use the method of elimination to eliminate one variable. Let's eliminate $x$ by adding equation $1$ and equation $2$ . $(x + 6y - z) + (-2x - 5y + 2z) = -5 + 3$ This simplifies to: $-x + y + z = -2$ Now we have the system: $1)\, -x + y + z = -2$ $2)\, -2x - 5y + 2z = 3$ $3)\, 5x - 4y + 5z = 19$
Eliminate Variable $x$ : Multiply equation $1$ by $2$ and add it to equation $2$ to eliminate $x$ . $2(-x + y + z) + (-2x - 5y + 2z) = 2(-2) + 3$ This simplifies to: $-2x + 2y + 2z - 2x - 5y + 2z = -4 + 3$ Combining like terms, we get: $-4x - 3y + 4z = -1$ Now we have the system: $1$ ) $-x + y + z = -2$ $2$ ) $-4x - 3y + 4z = -1$ $3$ ) $5x - 4y + 5z = 19$
Eliminate Variable x: Multiply equation $1$ by $5$ and add it to equation $3$ to eliminate x. $\newline$ $5(-x + y + z) + (5x - 4y + 5z) = 5(-2) + 19$ $\newline$ This simplifies to: $\newline$ $-5x + 5y + 5z + 5x - 4y + 5z = -10 + 19$ $\newline$ Combining like terms, we get: $\newline$ $y + 10z = 9$ $\newline$ Now we have the system: $\newline$ $1$ ) $-x + y + z = -2$ $\newline$ $2$ ) $-4x - 3y + 4z = -1$ $\newline$ $3$ ) $y + 10z = 9$
Solve for $y$ : Solve equation $3$ for $y$ . $y = 9 - 10z$ Now we can substitute this expression for $y$ into equations $1$ and $2$ .
Substitute y into Eq $1$ : Substitute $y = 9 - 10z$ into equation $1$ . $-x + (9 - 10z) + z = -2$ This simplifies to: $-x + 9 - 9z = -2$ Now solve for x: $-x = -2 - 9 + 9z$ $x = 11 - 9z$
Substitute $y$ into Eq $2$ : Substitute $y = 9 - 10z$ into equation $2$ . $\newline$ $-4x - 3(9 - 10z) + 4z = -1$ $\newline$ This simplifies to: $\newline$ $-4x - 27 + 30z + 4z = -1$ $\newline$ Now solve for $x$ : $\newline$ $-4x = -1 + 27 - 34z$ $\newline$ $x = (28 - 34z) / -4$ $\newline$ $x = -7 + (34/4)z$ $\newline$ $x = -7 + (17/2)z$
Solve for z: Equate the two expressions for $x$ and solve for $z$ . $11 - 9z = -7 + (\frac{17}{2})z$ Multiply through by $2$ to eliminate the fraction: $22 - 18z = -14 + 17z$ Combine like terms: $22 + 14 = 17z + 18z$ $36 = 35z$ Divide by $35$ : $z = \frac{36}{35}$ $z = 1.02857142857$ (approximately)
Substitute $z$ into $y$ : Substitute $z$ back into the expression for $y$ .
$y = 9 - 10z$
$y = 9 - 10(\frac{36}{35})$
$y = 9 - \frac{360}{35}$
$y = 9 - 10.2857142857$
$y = -1.2857142857$ (approximately)
Substitute $z$ into $x$ : Substitute $z$ back into the expression for $x$ .
$x = 11 - 9z$
$x = 11 - 9(\frac{36}{35})$
$x = 11 - \frac{324}{35}$
$x = 11 - 9.25714285714$
$x = 1.74285714286$ (approximately)