**Word problems using elimination method :**

In this section, we are going to see, how elimination method can be used to solve word problems on linear equations.

**Example 1 :**

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 no. of adult tickets and "y" be the no. of kids tickets.

According to the question, we have

x + y = 548 ---------(1)

And also,

10x + 5y = 3750

Divide both sides by 5.

2x + y = 750 --------(2)

**Step 2 : **

Eliminate one of the variables to get the value of the other variable.

In (1) and (2), variable "y" is having the same coefficient. But, the variable "y" is having the same sign in both the equations.

To change the sign of "y" in (1), multiply both sides of (1) by negative sign.

- (x + y) = - 548

- x - y = - 548 --------(3)

**Step 3 : **

Now, eliminate the variable "y"in (2) and (3) as given below and find the value of "x".

**Step 4 : **

Plug x = 202 in (1) to get the value of y.

(2) --------> 202 + y = 548

Subtract 202 from both sides.

aaaaaaaaaaaaaaaaaaaa 202 + y = 548 aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa- 202 - 202 aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa------------------- aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa y = 346 aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa------------------- aaaaaaaaaaaaaaaaa

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

**Example 2 :**

Sum of the cost price of two products is $50. Sum of the selling price of the same two products is $52. If one is sold at 20% profit and other one is sold at 20% loss, find the cost price of each product.

**Solution :**

**Step 1 :**

Let "x" and "y" be the cost prices of two products.

Then, x + y = 50 --------(1)

**Step 2 :**

Let us assume that "x" is sold at 20% profit

Then, the selling price of "x" is 120% of "x"

Selling price of "x" = 1.2x

Let us assume that "y" is sold at 20% loss

Then, the selling price of "y" is 80% of "y"

Selling price of "x" = 0.8y

Given : Selling price of "x" + Selling price of "y" = 52

1.2x + 0.8y = 52

To avoid decimal, multiply both sides by 10

12x + 8y = 520

Divide both sides by 4.

3x + 2y = 130 --------(2)

**Step 3 : **

Eliminate one of the variables to get the value of the other variable.

In (1) and (2), both the variables "x" and "y" are not having the same coefficient.

One of the variables must have the same coefficient.

So multiply both sides of (1) by 2 to make the coefficients of "y" same in both the equations.

(1) **⋅ **2 --------> 2x + 2y = 100 ----------(3)

Variable "y" is having the same sign in both (2) and (3).

To change the sign of "y" in (3), multiply both sides of (3) by negative sign.

- (2x + 2y) = - 100

- 2x - 2y = - 100 --------(4)

**Step 4 : **

Now, eliminate the variable "y"in (2) and (4) as given below and find the value of "x".

**Step 5 : **

Plug x = 30 in (1) to get the value of y.

(2) --------> 30 + y = 50

Subtract 30 from both sides.

aaaaaaaaaaaaaaaaaaaa 30 + y = 50 aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa - 30 - 30 aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa------------------- aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa y = 20 aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa------------------- aaaaaaaaaaaaaaaaa

Hence, the cost prices of two products are $30 and $20.

After having gone through the stuff given above, we hope that the students would have understood "Word problems using elimination method".

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