**Equations and relationships :**

We have to do the following three steps to solve any real world problems.

(i) Find the relationship between the quantities

(ii) Model the relationship as equation

(iii) Solve for the unknown quantity

For example,

The cost of each pen in a store is $0.75. Time would like to buy some pens. How much does Tim have to pay ?

To solve this problem, we have to assign variables for the unknown quantities "number of pens" and "Total money paid"

Let "x" be the number of pens and "y" be the total money paid for "x" number of pens.

Now, we have to find the relationship between the two quantities "x" and "y" and form it as an equation.

Since, each pen costs $0.75, the cost of "x" number of pens is

$0.75x.

Since "y" stands for the total money paid for "x" number of pens, we have

**y = $0.75x**

If Tim buys 4 pens, we have to plug x = 4 in the above equation to know the total money paid for 4 pens.

y = $0.75(4)

y = $3

Therefore, Tim has to pay $3 for 4 pens.

**Example 1 : **

Sonia used $12.50 to buy a new journal. She has $34.25 left in her savings account. How much money did Sonia have before she bought the journal ?

**Solution : **

She used $12.50 to buy journal and $34.25 left in her saving account.

If she had not bought journal, $12.50 would have been in her account.

To know the money that Sonia had before she bought the journal, we have to add $12.50 and $34.25

Then, we have

$12.50 + $34.25 = **$46.75**

Hence, Sonia had $46.75 before she bought the journal.

**Example 2 :**

Tim is paid $8 more than the number of bags of peanuts he sells at the baseball stadium. The table shows the relationship between the money Tim earns and the number of bags of peanuts Tim sells. Identify the independent and dependent variables, and write an equation that represents the relationship.

**Solution : **

The number of bags is the independent variable, and the money Tim earns is the dependent variable.

Since "x" stands for number of bags and it is given that he earns $8 more than number of bags (x), the money earned by Tim is

x + 8

Since "y" stands for money earned by Tim, we have

**y = x + 8**

Hence, the equation y = x + 8 expresses the relationship between the number of bags Tim sells and the amount he earns.

**Example 3 :**

The Falcons won their football game with a score of 30 to 19. Kevin scored 12 points for the Falcons. Write an equation to determine how many points Kevin’s teammates scored.

**Solution : **

Let "x" be the points scored by Kevin's team mates.

The points scored by Kevin is 12.

Kevin and team mates together won the game. So, the winning point is the sum of the points scored by Kevin and his team mates.

That is,

12 + x

Since the winning score is 30 points, we have

**12 + x = 30**

Hence, 12 + x = 30 is the equation to determine how many points Kevin’s teammates scored.

**Example 4 :**

A taxi charges a flat rate of $3, plus an additional $1.50 per mile. How can we model this situation as an equation ? And also find the fare for 10 miles of distance.

**Solution : **

Let "x" be the number of miles and "y" be the total fare for "x" number of miles.

Since taxi, charges $1.50 per mile, fare for "x" number of miles is

$1.50x

Taxi charges a flat rate of $3. Then the total fare is

$1.50x + $3

Since "y" stands for total fare for "x" number of miles, we have

**y = $1.50x + $3**

To find the fare for 15 miles of distance, plug x = 15 in the above equation.

y = $1.50(10) + $3

y = $15 + $3

**y = $18**

**Hence, the fare for 10 miles of distance is $18.**

After having gone through the stuff given above, we hope that the students would have understood "Equations and relationships".

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