For which of the following ordered pairs (µ, d), the system of linear equations x + 2y + 3z = 1 3x + 4y + 5z = µ 4x + 4y + 4z = d is inconsistent ?

Engineering

Mathematics

Solving System of Linear Equation Cramers Rule

Question

For which of the following ordered pairs (µ, d), the system of linear equations

x + 2y + 3z = 1

3x + 4y + 5z = µ

4x + 4y + 4z = d

is inconsistent ?

(4, 6)

(1, 0)

(3, 4)

(4, 3)

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Note $D = | \begin{matrix} 3 & 4 & 5 \\ 1 & 2 & 3 \\ 4 & 4 & 4 \end{matrix} | (R_{3} \to R_{3} - 2 R_{1} + 3 R_{2})$

$= | \begin{matrix} 3 & 4 & 5 \\ 1 & 2 & 3 \\ 0 & 0 & 0 \end{matrix} | = 0$

Now let P₃ $\equiv$ 4x + 4y + 4z – d = 0. If the system has solutions it will have infinite solution,

so P₃ $\equiv$ αP₁ + βP₂

Hence 3α + β = 4 & 4α + 2β = 4 ⇒ α = 2 & β = –2

So for infinite solution 2µ –2 = δ ⇒ for 2µ $\neq$ δ + 2

system inconsistent