Nguyen Savage
01/16/2024 · High School
Find the solution for \( x \), using determinants (also known as Cramer's Rule). Hint: No need to find solutions for \( y \) and \( z \). \[ \begin{array}{l}4 x-3 y+3 z=9 \\ 2 x-y+6 z=12 \\ x-4 y=1\end{array} \]
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Step-by-step Solution
Solve the system of equations \( 4x-3y+3z=9; 2x-y+6z=12; x-4y=1 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}4x-3y+3z=9\\2x-y+6z=12\\x-4y=1\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}4x-3y+3z=9\\y=-12+2x+6z\\x-4y=1\end{array}\right.\)
- step2: Substitute the value of \(y:\)
\(\left\{ \begin{array}{l}4x-3\left(-12+2x+6z\right)+3z=9\\x-4\left(-12+2x+6z\right)=1\end{array}\right.\)
- step3: Simplify:
\(\left\{ \begin{array}{l}-2x+36-15z=9\\-7x+48-24z=1\end{array}\right.\)
- step4: Solve the equation:
\(\left\{ \begin{array}{l}x=\frac{27-15z}{2}\\-7x+48-24z=1\end{array}\right.\)
- step5: Substitute the value of \(x:\)
\(-7\times \frac{27-15z}{2}+48-24z=1\)
- step6: Simplify:
\(-\frac{7\left(27-15z\right)}{2}+48-24z=1\)
- step7: Multiply both sides of the equation by LCD:
\(\left(-\frac{7\left(27-15z\right)}{2}+48-24z\right)\times 2=1\times 2\)
- step8: Simplify the equation:
\(-93+57z=2\)
- step9: Move the constant to the right side:
\(57z=2+93\)
- step10: Add the numbers:
\(57z=95\)
- step11: Divide both sides:
\(\frac{57z}{57}=\frac{95}{57}\)
- step12: Divide the numbers:
\(z=\frac{5}{3}\)
- step13: Substitute the value of \(z:\)
\(x=\frac{27-15\times \frac{5}{3}}{2}\)
- step14: Simplify:
\(x=1\)
- step15: Substitute the values of \(x,z:\)
\(y=-12+2\times 1+6\times \frac{5}{3}\)
- step16: Simplify:
\(y=0\)
- step17: Calculate:
\(\left\{ \begin{array}{l}x=1\\y=0\\z=\frac{5}{3}\end{array}\right.\)
- step18: Check the solution:
\(\left\{ \begin{array}{l}x=1\\y=0\\z=\frac{5}{3}\end{array}\right.\)
- step19: Rewrite:
\(\left(x,y,z\right) = \left(1,0,\frac{5}{3}\right)\)
The solution for the system of equations using Cramer's Rule is \( x = 1 \), \( y = 0 \), and \( z = \frac{5}{3} \) or \( z = 1.\dot{6} \).
Quick Answer
\( x = 1 \)
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