Solve the equations

We have,

…(i)

…(ii)

Substitute and in equations (i) and (ii), we get

5u – 2v = -1 …(iii)

15u + 7v = 10 …(iv)

In order to eliminate one of the variables (x and y), we need to make that variable’s coefficient equal in both of the equations.

In this question, we can eliminate v. Equation (iii) has a (-) sign before y and equation (iv) has a (+) sign before y.

Now, we need to make the coefficient of v equal, since, in equation (iii), v’s coefficient is 2 and in equation (iv), v’s coefficient is 7 (ignoring the signs before it).

For equal coefficient of v, multiply equation (iii) by 7 and equation (iv) by 2. (Multiplication has to be done over the whole equation as to balance the equation even after making changes)

So, we have

5u – 2v = -1 [× 7

15u + 7v = 10 [× 2

⇒ 35u – 14v = -7 …(v)

& 30u + 14v = 20 …(vi)

Now, we have equations (v) and (vi) which can be solved by eliminating variable v.

Recall equation (v) and (vi),

35u – 14v = -7

30u + 14v = 20

Solve these,

We get,

65u = 13

Put this values of x in equation (iii), we get

⇒ 1 – 2v = -1

⇒ 2v = 1 + 1

⇒ 2v = 2

⇒ v = 1

We know, and .

Or and .

⇒ x + y = 5 …(vii)

And

⇒ x – y = 1 …(viii)

Solving equations (vii) and (viii), we get

We get,

2x = 6

⇒ x = 3

Put this value of x in equation (viii), we get

(3) – y = 1

⇒ y = 3 – 1

⇒ y = 2

Thus, the solution is x = 3 and y = 2.