Elimination Method

STEP 1 : Write both the equations in the form ax+by=c.

STEP 2 : Multiply one or both equations so that the coefficients of either $x or y are equal but opposite in sign.$

STEP 3 : Add the equations to eliminate a variable.

STEP 4 : Solve the resulting equation for the variable.

STEP 5 : Substitute the value of the solved variable into one of the original equations to find the second variable.

Elimination method examples :

Solve the following pair of linear equations by the elimination method: x+y=5, 2x-3y=4

STEPS

(x+y=5)×3 = 3x+3y = 15 2x-3y= 4

5x = 19 ⇒ x = 19/5

x+y=5 ⇒ 19/5+y=5 ⇒ y=5 - 19/5 ⇒ y = 6/5

Answer: x= 1/5, y = 6/5

REASONING

Multiply eq(1) by 3

On adding , 3y and -3y get canceled.

Our GOAL is to obtain the coefficients that are equal but opposite in sign.

Plug in the value of x into eq(1) and solve for y

Solve the following pair of linear equations by the elimination method: 3x-5y-4=0, 9x=2y+7

STEPS

3x-5y=4 9x-2y=7

(3x-5y=4)×(-3) = -9x+15y = -12 9x - 2y = 7

13y = -5 ⇒ y = -5/13

3x-5y=4 ⇒ 3x-5(-5/13) =4 ⇒ 3x = 4-25/13 ⇒ x = 9/13

Answer: x= 9/13, y = -5/13

REASONING

Setup the equations in the form ax+by=c

Multiply eq(1) by -3. The coefficents of x are 9 and 3 and 9 is divisible by 3.

So, we need to multiply only one eqution.

Plug in the value of x into eq(1) and solve for y

Solve the following pair of linear equations by the elimination method: x/2+2y/3=-1, x-y/3=3

Solve the following pair of linear equations by the elimination method: 2/x+3/y=13, 5/x-4/y=-2

FAQs

When should I use the elimination method?

The elimination method is most effective when:

The coefficients of one variable are easily aligned.

There are no fractions or complex terms, making it easier to eliminate variables.

What if the elimination method results in a false statement (like 0=5)

If the elimination method leads to a false statement it means the system of equations has no solution and the lines are parallel.

What if the elimination method results in a true statement (like 0=0)? $If you get a true statement the system has infinitely many solutions, and the two equations represent the same line.$

What’s the difference between the substitution and elimination methods?

In the substitution method, one equation is solved for one variable and then substituted into the other equation. In the elimination method, the equations are added or subtracted to eliminate one variable directly. Elimination is often faster for systems where the coefficients are already aligned or can be easily aligned.

Hem Chandra Bhatt