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. 

Examples

Example 1 :

Solve by elimination method. 

3x + 4y  =  -25

2x - 3y  =  6

Solution :

3x + 4y  =  -25 ---- (1)

2x - 3y  =  6  ---- (2)

Both x terms and y terms have different coefficients in the above system of equations.

Let's try to make the coefficients of y terms equal.

To make the coefficients of y terms equal, we have to find the least common multiple 4 and 3.

The least common multiple of 4 and 3 is 12.

Multiply the first equation by 3 in order to make the coefficient of y as 12 and multiply the second equation by 4 in order to make the coefficient of y as -12. 

(1) ⋅ 3 ---->  9x + 12y  =  -75

(2) ⋅ 4 ---->  8x - 12y  =  24

Now, we can add the two equations and eliminate y as shown below. 

Divide each side by 17. 

x  =  -3

Substitute -3 for x in (1). 

(1)---->  3(-3) + 4y  =  -25

-9 + 4y  =  -25

Add 9 to each side.

4y  =  -16

Divide each side by 4.

y  =  -4

So, the values of x and y are -3 and -4 respectively.

Example 2 :

Solve by elimination method

2x + 3y  =  5

3x + 4y  =  7

Solution :

2x + 3y  =  5  ----(1)

3x + 4y  =  7  ----(2)

Both x terms and y terms have different coefficients in the above system of equations.

Let's try to make the coefficients of x terms equal.

To make the coefficients of y terms equal, we have to find the least common multiple 2 and 3.

The least common multiple of 2 and 3 is 6.

Multiply the first equation by 3 in order to make the coefficient of x as 6 and multiply the second equation by -2 in order to make the coefficient of x as -6. 

(1) ⋅ 3 ---->  6x + 9y  =  15

(2) ⋅ -2 ----> -6x - 8y  =  -14

Now, we can add the two equations and eliminate x as shown below. 

Substitute 1 for y in (1). 

(1)---->  2x + 3(1)  =  5

2x + 3  =  5

Subtract 3 from each side. 

2x  =  2

Divide each side by 2.

x  =  1

So, the values of x and y are 1 and 1 respectively.

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  =  306

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

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