System of linear equations

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Question

If the system of equations $x + y + z = 5, x + 2 y + 3 z = 9, x + 3 y + α z = β$ has infinitely many solutions, then $β^{3} - α^{3} =$

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Solution

Augmented matrix $= [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix} \begin{matrix} 1 & 1 & 5 \\ 2 & 3 & 9 \\ 3 & α & β \end{matrix}] R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - R_{1}$
$\begin{array}{l} ~ [\begin{matrix} 1 & 1 & 1 & 5 \\ 0 & 1 & 2 & 4 \\ 0 & 2 & α - 1 & β - 5 \end{matrix}] R_{3} \to R_{3} - 2 R_{2} \\ ~ [\begin{matrix} 1 & 1 & 1 & 5 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & α - 5 & β - 13 \end{matrix}] \end{array}$
$S y s t e m h a s m a n y s o l u t i o n s \Rightarrow α - 5 = 0 a n d β - 13 = 0$ .
$\Rightarrow α = 5, β = 13$
$∴ β^{3} - α^{3} = 13^{3} - 5^{3} = 2072$

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System of linear equations

Remember concepts with our Masterclasses.

If the system of equations x+y+z=5, x+2y+3z=9, x+3y+αz=β has infinitely many solutions, then β3−α3=

Augmented matrix =111 1152393αβ R2→R2−R1, R3→R3−R1 ~1115012402α-1β-5 R3→R3-2R2~1115012400α-5β-13System has many solutions⇒α-5=0 and β-13=0.⇒ α = 5,β=13∴ β3−α3=133−53=2072

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If the system of equations $x + y + z = 5, x + 2 y + 3 z = 9, x + 3 y + α z = β$ has infinitely many solutions, then $β^{3} - α^{3} =$