Question

Find the value of x + y and x - y if 27x + 31y = 85 and 31x + 27y = 89

Answer 1

I Hope It Will Help You ....

Answer 2

Answer:

The required values are

$\boxed{\red{\sf\:x\:+\:y\:=\:3}}\sf\:\:\:\&\:\:\:\boxed{\red{\sf\:x\:-\:y\:=\:1}}$

Step-by-step-explanation:

The given simultaneous equations are

$\sf\:27x\:+\:31y\:=\:85\:\:\:-\:-\:(\:1\:)\\\\\sf\:31x\:+\:27y\:=\:89\:\:\:-\:-\:(\:2\:)$

By adding equation ( 1 ) and equation ( 2 ), we get,

$\sf\:27x\:+\:31y\:=\:85\:\:\:-\:-\:(\:1\:)\\\sf\:+\\\underline{\sf\:31x\:+\:27y\:=\:89}\sf\:\:\:-\:-\:(\:2\:)\\\\\\\implies\sf\:58x\:+\:58y\:=\:174\\\\\\\implies\boxed{\red{\sf\:x\:+\:y\:=\:3}}\sf\:\:-\:-\:[\:Dividing\:by\:58\:]$

Now, by subtracting equation ( 1 ) from equation ( 2 ), we get,

$\sf\:31x\:+\:27y\:=\:89\:\:\:-\:-\:(\:2\:)\\\sf\:-\\\underline{\sf\:27x\:+\:31y\:=\:85}\sf\:\:\:-\:-\:(\:1\:)\\\\\\\implies\sf\:4x\:-\:4y\:=\:4\\\\\\\implies\boxed{\red{\sf\:x\:-\:y\:=\:1}}\sf\:\:-\:-\:[\:Dividing\:by\:4\:]$

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.

3. Trick to solve problems like x + y & x - y:

In some linear equations, the coefficients of x and y are interchanged in the two equations.

In this case, we use following steps:

1. Add both the equations.

2. Reduce the obtained equation to x + y.

3. Subtract the second equation from the first ( or first from the second ).

4. Reduce the obtained equation to x - y.

To get the solution of the given simultaneous equations, solve both the new equations i. e. x + y & x - y.