Question

Find the value of p for which the system of equations 2x + 3y = 5 and px + 2y = 6 has a
unique solution.

Answer 1

Answer:

2x + 3y = 5

x = (5 - 3y)/2

px + 2y = 6

p(5-3y)/2 +2y =6

p( 5 - 3y) + 4y = 12

p( 5 - 3y) =12 - 4y

p = 12-4y/5 - 3y

Step-by-step explanation:

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Answer 2

Answer:

The value of p is any real number other than $\displaystyle{\sf\:\dfrac{4}{3}}$.

Step-by-step-explanation:

The given simultaneous equations are

2x + 3y = 5 - - - ( 1 )

px + 2y = 6 - - - ( 2 )

Comparing equation ( 1 ) with ax + by + c = 0, we get,

2x + 3y = 5 - - - ( 1 )

• a₁ = 2
• b₁ = 3
• c₁ = 5

Comparing equation ( 2 ) with ax + by + c = 0, we get,

px + 2y = 6 - - - ( 2 )

• a₂ = p
• b₂ = 2
• c₂ = 6

Now, we know that,

The condition for unique solution of simultaneous equations is

$\displaystyle{\pink{\sf\:\dfrac{a_1}{a_2}\:\ne\:\dfrac{b_1}{b_2}}}$

$\displaystyle{\implies\sf\:\dfrac{2}{p}\:\ne\:\dfrac{3}{2}}$

$\displaystyle{\implies\sf\:2\:\times\:2\:\ne\:3\:\times\:p}$

$\displaystyle{\implies\sf\:4\:\ne\:3\:p}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:p\:\ne\:\dfrac{4}{3}}}}}$

∴ The value of p is any real number other than $\displaystyle{\sf\:\dfrac{4}{3}}$ for which the given system of equations has unique solution.