ELIMINATION METHOD

The flow chart shown below explains the steps to be done in solving system of linear equations with two unknowns 'x' and 'y' using elimination method. 

Example 1 :

Solve by elimination method. 

3x + 4y  =  -25

2x - 3y  =  6

Solution :

3x + 4y  =  -25 ---- (1)

2x - 3y  =  6  ---- (2)

Since the coefficients are not same, we can try to make them same by taking LCM.

Eliminating y  and solve for x :

LCM (3, 4) = 12

(1) ⋅ 3 ==> 9x + 12y = -75

(2) ⋅ 4 ==> 8x - 12y = 24

By observing the equations now, they are having different signs. So, we have to add them.

9x + 12y + 8x - 12y = -75 + 24

17x = -51

Dividing by 17 on both sides.

x = -51/17

x = -3

Finding the value of y :

By applying the value of x in (2), we get

2(-3) - 3y = 6

-6 - 3y = 6

Add 6 on both sides, we get

-3y = 12

Dividing by -3 on both sides.

y = -4

Example 2 :

Solve by elimination method

2x + 3y  =  5

3x + 4y  =  7

Solution :

2x + 3y  =  5  ----(1)

3x + 4y  =  7  ----(2)

Since the coefficients are not same, we can try to make them same by taking LCM.

Eliminating y and solve for x :

LCM (2, 3) = 6

(1) ⋅ 3 ==> 6x + 9y = 15

(2) ⋅ 2 ==> 6x + 8y = 14

By observing the equations now, they are having same signs. So, we have to subtract them.

6x + 9y - 6x - 8y = 15 - 14

x = 1

Finding the value of y :

By applying the value of x in (1), we get

2(1) + 3y = 5

2 + 3y = 5 

Subtracting 2 on both sides.

3y = 3

Dividing by 3 on both sides.

y = 1

Example 3 : 

A park charges $10 for adults and $5 for kids. How many many adults tickets and kids tickets were sold, if a total of 548 tickets were sold for a total of $3750 ? 

Solution :

Step 1 :

Let "x" be the number of adults tickets and "y" be the number of kids tickets.

No. of adults tickets + No. of kids tickets  =  Total

x + y  =  548 ----(1)

Step 2 :

Write an equation which represents the total cost.

Cost of x no. adults tickets  =  10x

Cost of y no. of kids tickets  =  5y

Total cost  =  $3750

Then, we have 

10x + 5y  =  3750

Divide both sides by 5.

2x + y  =  750 ----(2)

Step 3 :

Solve (1) and (2) using elimination method. 

x + y  =  548 ----(1)

2x + y  =  750 ----(2)

In the above two equations, y is having the same coefficient, that is 1. 

Multiply the first equation by -1 to get the coefficient of -1. And keep the second equation as it is. 

Then, we have 

-x - y  =  - 548

2x + y  =  750

We can add the above two equations and eliminate y. 

Then, we have

x  =  202

Step 4 : 

Substitute 202 for x in the first equation.

(1)---->  202 + y  =  548

Subtract 202 from each side.

y  =  346

So, the number of adults tickets sold is 202 and the number of kids tickets sold is 346.

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