Question

solve the following equation by elimination method:
x + 2y = 1 , 3x - y= 17​

Answer 1

Answer:

Solution:

The standard form of two linear simultaneous equations in two unknowns are:

a₁x + b₁y + c₁ = 0 ………………………………………………….…………………(1)

a₂x + b₂y + c₂ = 0 …………………………………………………………….………(2)

There are more than one method to solve the above types of equations. But the most general is the cross-multiplication method. According to the rule of cross-multiplication, the solutions are given by the equality:

x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - a₁c₂) = 1/(a₁b₂ - a₂b₁) ………………………….(α)

While apply the relations (α) to find x and y for equations (1) and (2), one has to bear in mind the following:

(i) Always start with the y-coefficient.

(ii) Descend first and then ascend.

(iii) Move to the right having started with y-term till you exhaust the cycle.

(iv) Any product formed by descending is positive and any product formed by ascending is negative.

If you follow the above rule, you will get the three denominators for x, y, and the constant term, whence x and y can be obtained easily.

The equations under investigation are

1/x + 1/y = 7 ……………………………………………………………………………..(3)

2/x + 3/y = 17 ……………………………………………………………………………(4)

To bring (3) and (4) to the form (1) and (2), we put

X = 1/x and Y = 1/y …………………………………………………………….…….(5)

Then (3) and (4) take the standard form

X + Y - 7 = 0

2X + 3Y - 17 = 0

Comparing the above two simultaneous equations in X and Y with (1) and (2), we get

a₁ = 1, b₁= 1, c₁= -7 ; a₂ = 2, b₂ = 3, c₂ = -17

Substituting the above values for the coefficients in (α),

X/[1.(-17) - 3.(-7)] = Y/[(-7).2 - 1.(-17)] = 1/(1.3 - 2.1)

Or, X/(-17 + 21) = Y/(-14 + 17) = 1/(3 - 2)

Or, X/4 = Y/3 = 1/1

⇒ X = 4/1 = 4 and Y = 3/1 = 3

To revert to the original variables x and y, we use the relation (5).

1/x = 4 ⇒ x = 1/4

1/y = 3 ⇒ y = 1/3

Hence the solutions are x = 1/4 and y = 1/3 (Proved)