Gibbs Evans
06/30/2024 · Junior High School
Solve the system \[ \begin{array}{l}5 x+3 y-5 z=-34 \\ 3 x-4 y+3 z=-14 \\ 6 x-3 y+6 z=-18\end{array} \]
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Step-by-step Solution
Solve the system of equations \( 5x+3y-5z=-34;3x-4y+3z=-14;6x-3y+6z=-18 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}5x+3y-5z=-34\\3x-4y+3z=-14\\6x-3y+6z=-18\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}x=\frac{-34-3y+5z}{5}\\3x-4y+3z=-14\\6x-3y+6z=-18\end{array}\right.\)
- step2: Substitute the value of \(x:\)
\(\left\{ \begin{array}{l}3\times \frac{-34-3y+5z}{5}-4y+3z=-14\\6\times \frac{-34-3y+5z}{5}-3y+6z=-18\end{array}\right.\)
- step3: Simplify:
\(\left\{ \begin{array}{l}\frac{3\left(-34-3y+5z\right)}{5}-4y+3z=-14\\\frac{6\left(-34-3y+5z\right)}{5}-3y+6z=-18\end{array}\right.\)
- step4: Solve the equation:
\(\left\{ \begin{array}{l}y=\frac{-32+30z}{29}\\\frac{6\left(-34-3y+5z\right)}{5}-3y+6z=-18\end{array}\right.\)
- step5: Substitute the value of \(y:\)
\(\frac{6\left(-34-3\times \frac{-32+30z}{29}+5z\right)}{5}-3\times \frac{-32+30z}{29}+6z=-18\)
- step6: Simplify:
\(-\frac{6\left(178-11z\right)}{29}-\frac{3\left(-32+30z\right)}{29}+6z=-18\)
- step7: Multiply both sides of the equation by LCD:
\(\left(-\frac{6\left(178-11z\right)}{29}-\frac{3\left(-32+30z\right)}{29}+6z\right)\times 29=-18\times 29\)
- step8: Simplify the equation:
\(-972+150z=-522\)
- step9: Move the constant to the right side:
\(150z=-522+972\)
- step10: Add the numbers:
\(150z=450\)
- step11: Divide both sides:
\(\frac{150z}{150}=\frac{450}{150}\)
- step12: Divide the numbers:
\(z=3\)
- step13: Substitute the value of \(z:\)
\(y=\frac{-32+30\times 3}{29}\)
- step14: Simplify:
\(y=2\)
- step15: Substitute the values of \(y,z:\)
\(x=\frac{-34-3\times 2+5\times 3}{5}\)
- step16: Simplify:
\(x=-5\)
- step17: Calculate:
\(\left\{ \begin{array}{l}x=-5\\y=2\\z=3\end{array}\right.\)
- step18: Check the solution:
\(\left\{ \begin{array}{l}x=-5\\y=2\\z=3\end{array}\right.\)
- step19: Rewrite:
\(\left(x,y,z\right) = \left(-5,2,3\right)\)
The solution to the system of equations is \( x = -5, y = 2, z = 3 \).
Quick Answer
\( x = -5, y = 2, z = 3 \).
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