Question

5/x+y -2/x-y=-1
15/x+y +7/x-y= 10
​

Answer 1

Answer:

The solution of the given simultaneous equations is

$\boxed{\red{\sf\:x\:=\:3}}\sf\:\:\&\:\:\boxed{\red{\sf\:y\:=\:2}}$

Step-by-step-explanation:

The given simultaneous equations are

$\sf\:\dfrac{5}{x\:+\:y}\:-\:\dfrac{2}{x\:-\:y}\:=\:-\:1\\\\\\\sf\:\dfrac{15}{x\:+\:y}\:+\:\dfrac{7}{x\:-\:y}\:=\:10$

By replacing $\sf\:\dfrac{1}{x\:+\:y}$ with $\sf\:a$ and $\sf\:\dfrac{1}{x\:-\:y}$ with $\sf\:b$ we get,

$\sf\:5a\:-\:2b\:=\:-\:1\:\:\:-\:-\:(\:1\:)\\\\\\\sf\:15a\:+\:7b\:=\:10\:\:\:-\:-\:(\:2\:)$

By multiplying equation ( 1 ) by 3, we get,

$\sf\:15a\:-\:6b\:=\:-\:3\:\:\:-\:-\:(\:3\:)$

By subtracting equation ( 3 ) from equation (2), we get,

$\sf\:\cancel{15a}\:+\:7b\:=\:10\:\:\:-\:-\:(\:2\:)\\\sf\:-\\\underline{\sf\:\cancel{15a}\:-\:6b\:=\:-\:3}\sf\:\:\:-\:-\:(\:3\:)\\\\\\\implies\sf\:13b\:=\:13\\\\\\\implies\sf\:b\:=\:\cancel{\frac{13}{13}}\\\\\\\implies\boxed{\red{\sf\:b\:=\:1}}$

By substituting b = 1 in equation ( 1 ), we get,

$\sf\:5a\:-\:2b\:=\:-\:1\:\:\:-\:-\:(\:1\:)\\\\\\\implies\sf\:5a\:-\:2\:\times\:1\:=\:-\:1\\\\\\\implies\sf\:5a\:=\:-\:1\:+\:2\\\\\\\implies\sf\:5a\:=\:1\\\\\\\implies\boxed{\red{\sf\:a\:=\:\frac{1}{5}}}$

Now, by replacing again $\sf\:\dfrac{1}{x\:+\:y}$ with $\sf\:a$, we get,

$\sf\:\dfrac{1}{x\:+\:5}\:=\:a\\\\\\\implies\sf\:\dfrac{1}{x\:+\:5}\:=\:\dfrac{1}{5}\\\\\\\implies\sf\:x\:+\:y\:=\:5\:\:-\:-\:-\:(\:4\:)$

Now, by again replacing $\sf\:\dfrac{1}{x\:-\:y}$ with $\sf\:b$ , we get,

$\sf\:\dfrac{1}{x\:-\:y}\:=\:b\\\\\\\implies\sf\:\dfrac{1}{x\:-\:y}\:=\:1\\\\\\\implies\sf\:x\:-\:y\:=\:1\:\:\:-\:-\:(\:5\:)$

By adding equation ( 4 ) and equation ( 5 ), we get,

$\sf\:x\:+\:\cancel{y}\:=\:5\:\:\:-\:-\:(\:4\:)\\+\\\underline{\sf\:x\:-\:\cancel{y}\:=\:1}\sf\:\:\:-\:-\:(\:5\:)\\\\\\\implies\sf\:2x\:=\:6\\\\\\\implies\sf\:x\:=\:\cancel{\frac{6}{2}}\\\\\\\implies\boxed{\red{\sf\:x\:=\:3}}$

By substituting x = 3 in equation ( 4 ), we get,

$\sf\:x\:+\:y\:=\:5\:\:\:-\:-\:(\:4\:)\\\\\\\implies\sf\:3\:+\:y\:=\:5\\\\\\\implies\sf\:y\:=\:5\:-\:3\\\\\\\implies\boxed{\red{\sf\:y\:=\:2}}$

Additional Information:

1. Linear Equations in two variables:

The equation with the highest index (degree) 1 is called as linear equation. If the equation has two different variables, it is called as 'linear equation in two variables'.

The general formula of linear equation in two variables is

ax + by + c = 0

Where, a, b, c are real numbers and

a ≠ 0, b ≠ 0.

2. Solution of a Linear Equation:

The value of the given variable in the given linear equation is called the solution of the linear equation.

Answer 2

Answer:

The solution of the given simultaneous equations is

Step-by-step-explanation:

The given simultaneous equations are

By replacing  with  and  with  we get,

By multiplying equation ( 1 ) by 3, we get,

By subtracting equation ( 3 ) from equation (2), we get,

By substituting b = 1 in equation ( 1 ), we get,

Now, by replacing again  with , we get,

Now, by again replacing  with  , we get,

By adding equation ( 4 ) and equation ( 5 ), we get,

By substituting x = 3 in equation ( 4 ), we get,

Additional Information:

1. Linear Equations in two variables:

The equation with the highest index (degree) 1 is called as linear equation. If the equation has two different variables, it is called as 'linear equation in two variables'.

The general formula of linear equation in two variables is

ax + by + c = 0

Where, a, b, c are real numbers and

a ≠ 0, b ≠ 0.

2. Solution of a Linear Equation:

The value of the given variable in the given linear equation is called the solution of the linear equation