Question

solve the following pair of linear equations by the substitution method : (1)2x +y =5,4x+6y=14 (2)x+y=4,x-3y=4 (3)3x-5y-4=0, 9x-2y-7=0​

Answer 1

Required Solutions:-

(1) Given equations:-

• 2x + y = 5
• 4x + 6y = 14

To find:-

• The value of x and y

Method:-

• Substitution method.

Solution:-

$\sf{2x + y = 5 \longrightarrow[i]}$

$\sf{4x + 6y = 14 \longrightarrow[ii]}$

From eq.[i]

$\sf{2x + y = 5}$

= $\sf{y = 5-2x}$

Substituting the value of y in equation (ii)

$\sf{4x + 6y = 14}$

= $\sf{4x + 6(5-2x) = 14}$

= $\sf{4x + 30 - 12x = 14}$

= $\sf{30-8x = 14}$

= $\sf{-8x = 14-30}$

= $\sf{-8x = -16}$

= $\sf{x = \dfrac{-16}{-8}}$

=> $\sf{x = 2}$

Putting the value of x in equation (i)

$\sf{2x + y = 5}$

= $\sf{2\times 2 + y = 5}$

= $\sf{4 + y = 5}$

=> $\sf{y = 5-4}$

=> $\sf{y = 1}$

Therefore,

• x = 2
• y = 1

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(2) Given equations:-

• x + y = 4
• x - 3y = 4

To find:-

• The value of x and y

Method:-

• Substitution method

Solution:-

$\sf{x + y = 4\longrightarrow[i]}$

$\sf{x-3y = 4\longrightarrow[ii]}$

From equation(i)

$\sf{x + y = 4}$

= $\sf{x = 4-y}$

Substituting the value of x in equation(ii)

$\sf{x-3y = 4}$

= $\sf{4 - y - 3y = 4}$

= $\sf{4-4y=4}$

= $\sf{-4y = 4-4}$

= $\sf{-4y = 0}$

= $\sf{y = \dfrac{0}{-4}}$

= $\sf{y = 0}$

Putting the value of y in equation(i)

$\sf{x + y = 4}$

= $\sf{x + 0 = 4}$

= $\sf{x = 4-0}$

= $\sf{x = 4}$

Therefore,

• x = 4
• y = 0

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(3) Given equations:-

• 3x - 5y - 4 = 0
• 9x - 2y - 7 = 0

To Find:-

• The value of x and y

Method:-

• Substitution method

Solution:-

From given:-

$\sf{3x - 5y - 4 = 0}$

=> $\sf{3x-5y=0\longrightarrow[i]}$

$\sf{9x - 2y - 7 = 0}$

=> $\sf{9x - 2y = 7\longrightarrow[ii]}$

From equation(i)

$\sf{3x - 5y = 4}$

= $\sf{3x = 4 +5y}$

=> $\sf{x = \dfrac{4+5y}{3}}$

Substituting the value of x in equation(ii)

$\sf{9x - 2y = 7}$

= $\sf{9\bigg(\dfrac{4+5y}{3}\bigg) - 2y = 7}$

= $\sf{3(4+5y) - 2y = 7}$

= $\sf{12 + 15y - 2y = 7}$

= $\sf{13y = 7-12}$

= $\sf{y = \dfrac{-5}{13}}$

Putting the value of y in equation(i)

$\sf{3x - 5y = 4}$

= $\sf{3x -5\bigg(\dfrac{-5}{13}\bigg) = 4}$

= $\sf{3x + \dfrac{25}{13} = 4}$

= $\sf{3x = 4 - \dfrac{25}{13}}$

= $\sf{3x = \dfrac{52-25}{13}}$

= $\sf{3x = \dfrac{27}{13}}$

=> $\sf{x = \dfrac{27}{13\times 3}}$

=> $\sf{x = \dfrac{9}{13}}$

Therefore,

• x = 9/13
• y = -5/13

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Answer 2

Answer:

$\Large\underline{\underline{\sf{ \color{magenta}{\qquad Given \qquad} }}}$

1)2x +y =5,4x+6y=14

Answer:-

2x − 3y = 7

2x − 3y = 7− 4x + 6y = 14

Explanation:-

Given System: 2x − 3y = 7...... (1)

− 4x + 6y = 14 ...... (2)

Step 1: We use elimination method to solve the given system. Adding the two equations gives

-2x+3y=21,

which does not eliminate either variable. However, we can multiply each equation by a suitable number so that the coefficients of one of the variables are opposites.

Step 2: To eliminate y multiply each side of equation (1) by 2 to get,

4x-6y=14

-4x+6y=15

Adding in columms we get:-

0=0

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2) x+y=4,x-3y=4

The solution for the system of equations is

x=−2,y=2

Explanation:-

$x - y = - 4$

$x = - 4 + 4y$

$x + 3y = 4$

Substituting equation 1 in 2 to find x

x + 3y = 4

−4 + y + 3y = 4

−4 + 4y = 4

4y = 8

y = 2

Substituting y in equation 1 to obtain x :

x = -4 + y

x = - 4 + 2

x = -2

Answer ⤵️

x = -2