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)

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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