Determine the value of p for which the system px + 3y = p-3 12x py p is inconsistent

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$\esone{p x + 3 y = [p - 3]} $

\left\{\begin{matrix}p=-\frac{3\left[y+1\right]}{x-1}\text{, }&x\neq 1\\p\in \mathrm{R}\text{, }&y=-1\text{ and }x=1\end{matrix}\right.

\left\{\begin{matrix}x=-\frac{3y-p+3}{p}\text{, }&p\neq 0\\x\in \mathrm{R}\text{, }&p=0\text{ and }y=-1\end{matrix}\right.

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px+3y-p=-3

Subtract p from both sides.

px-p=-3-3y

Subtract 3y from both sides.

\left[x-1\right]p=-3-3y

Combine all terms containing p.

\left[x-1\right]p=-3y-3

The equation is in standard form.

\frac{\left[x-1\right]p}{x-1}=\frac{-3y-3}{x-1}

Divide both sides by x-1.

p=\frac{-3y-3}{x-1}

Dividing by x-1 undoes the multiplication by x-1.

p=-\frac{3\left[y+1\right]}{x-1}

Divide -3-3y by x-1.

px=p-3-3y

Subtract 3y from both sides.

px=-3y+p-3

The equation is in standard form.

\frac{px}{p}=\frac{-3y+p-3}{p}

Divide both sides by p.

x=\frac{-3y+p-3}{p}

Dividing by p undoes the multiplication by p.

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Question 12 - CBSE Class 10 Sample Paper for 2019 Boards - Solutions of Sample Papers for Class 10 Boards

Last updated at Sept. 24, 2021 by

Question 12

For what value of p will the following pair of linear equations have infinitely many solutions

[p – 3]x + 3y = p

px + py = 12

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Transcript

Question 12 For what value of p will the following pair of linear equations have infinitely many solutions [p – 3]x + 3y = p px + py = 12 Given equations [p – 3]x + 3y = p px + py = 12 [p – 3]x + 3y = p [p – 3]x + 3y – p = 0 Comparing with a1x + b1y + c1 = 0 ∴ a1 = [p – 3] , b1 = 3 , c1 = –p px + py = 12 px + py – 12 = 0 Comparing with a2x + b2y + c2 = 0 ∴ a2 = p , b2 = p , c2 = –12 Given that Equation has infinite number of solutions ∴ 𝑎1/𝑎2 = 𝑏1/𝑏2 = 𝑐1/𝑐2 Putting in values [[𝑝 − 3]]/𝑝 = 3/𝑝 = [−𝑝]/[−12] [[𝑝 − 3]]/𝑝 = 3/𝑝 = 𝑝/12 1/2 marks Solving [[𝒑 − 𝟑]]/𝒑 = 𝟑/𝒑 p[p – 3] = 3p p2 – 3p = 3p p2 – 3p – 3p = 0 p2 – 6p = 0 p[p – 6] = 0 So, p = 0, 6 1/2 marks Solving 𝟑/𝒑 = 𝒑/𝟏𝟐 3 × 12 = p2 36 = p2 p2 = 36 p = ± √36 p = ± 6 So, p = 6, –6 1/2 marks Since p = 6 satisfies both equations. Hence, p = 6 is the answer 1/2 marks

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