Question

Solve the given equations by cramer’s rule. 3x-y=5, x+3y=5

Answer 1

Answer:

The solution of the given simultaneous equations is ( x, y ) = ( 1, 2 ).

Step-by-step-explanation:

The given simultaneous equations are

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

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

Now, comparing equation ( 1 ) with ax + by = c, we get,

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

• a₁ = 3
• b₁ = - 1
• c₁ = 5

Now, comparing equation ( 2 ) with ax + by = c, we get,

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

• a₂ = 1
• b₂ = 3
• c₂ = 5

Now, we know that,

D = $\displaystyle{\bigg|\sf\:a_1 \:\:\: b_1 \:\bigg|}$

$\displaystyle{\:\:}$ $\displaystyle{\quad\bigg|\sf\: a_2 \:\:\:b_2 \:\bigg|\quad}$

⇒ D = $\displaystyle{\bigg|\sf\:3 \:\:\:- 1 \:\bigg|}$

$\displaystyle{\qquad}$$\displaystyle{\quad\bigg|\sf\:1 \:\:\: 3 \:\bigg|\quad}$

⇒ D = 3 * 3 - [ ( - 1 ) * 1 ]

⇒ D = 9 - ( - 1 )

⇒ D = 9 + 1

⇒ D = 10

Now, we know that,

Dₓ = $\displaystyle{\bigg|\sf\: c_1\:\:\: b_1\:\bigg|}$

$\displaystyle{\:\:}$ $\displaystyle{\quad\bigg|\sf\:c_2 \:\:\:b_2\:\bigg|\quad}$

⇒ Dₓ = $\displaystyle{\bigg|\sf\:5 \:\:\: - 1\:\bigg|}$

$\displaystyle{\qquad}$$\displaystyle{\quad\bigg|\sf\: 5 \:\:\: 3 \:\bigg|\quad}$

⇒ Dₓ = 5 * 3 - [ ( - 1 ) * 5 ]

⇒ Dₓ = 15 - ( - 5 )

⇒ Dₓ = 15 + 5

⇒ Dₓ = 20

Now, we know that,

Dʸ = $\displaystyle{\bigg|\sf\:a_1 \:\:\:c_1\:\bigg|}$

$\displaystyle{\:\:}$ $\displaystyle{\quad\bigg|\sf\: a_2 \:\:\: c_2\:\bigg|\quad}$

⇒ Dʸ = $\displaystyle{\bigg|\sf\:3\:\:\:5\:\bigg|}$

$\displaystyle{\qquad}$ $\displaystyle{\quad\bigg|\sf\:1 \:\:\: 5\:\bigg|\quad}$

⇒ Dʸ = 3 * 5 - ( 5 * 1 )

⇒ Dʸ = 15 - 5

⇒ Dʸ = 10

Now, by Cramer's rule,

x = Dₓ / D

⇒ x = 20 / 10

⇒ x = 2

Now, by Cramer's rule,

y = Dʸ / D

⇒ y = 10 / 10

⇒ y = 1

∴ The solution of the given simultaneous equations is ( x, y ) = ( 1, 2 ).