**Solving an equation with variables on both sides :**

Equations with variables on both sides can be used to compare costs of real-world situations. To solve these equations, use inverse operations to get the variable terms on one side of the equation.

**Step 1 : **

To solve the equations with variables on both sides, we have to get rid of the variable on one of the sides using addition/subtraction.

**Step 2 : **

After getting the variable on only one side of the equation, isolate the variable on that side using the binary operations (addition, subtraction, multiplication and division) to get of the other terms that we have with the variable.

**Example 1 :**

Solve for x :

3x - 1 = x + 5

**Solution : **

**Step 1 : **

To get rid of "x" on the right side, we have to subtract "x" on both sides.

aaaaaaaaaaaaaaaaaaaa3x - 1 = x + 5 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa -x -x aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa---------------- aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa 2x - 1 = 5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa----------------

**Step 2 : **

To get rid "-1", add "1" to both sides of the equation.

aaaaaaaaaaaaaaaaaaaa2x - 1 = 5 aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaa + 1 = +1 aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa-------------- aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa 2x = 6 aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa--------------

**Step 3 : **

To get rid of "2" multiplied by x, divide by 2 on both sides.

2x / 2 = 6 / 2

x = 3

**Example 2 :**

David's Rental Car charges an initial fee of $20 plus an additional $30 per day to rent a car. Alex's Rental Car charges an initial fee of $36 plus an additional $28 per day. For what number of days is the total cost charged by both of them the same ?

**Solution : **

**Let "x" be the number of days for which the total cost charged by both of them is same.**

**Step 1 : **

Write an expression using "x" representing the total cost of renting a car from David’s Rental Car.

Total cost = Initial fee + cost for "x" days

Total days = 20 + 30x

**Step 2 : **

Write an expression using "x" representing the total cost of renting a car from Alex’s Rental Car.

Total cost = Initial fee + cost for "x" days

Total days = 36 + 28x

**Step 3 : **

We have assumed that the total cost charged by both of them is same for "x" number of days.

So, we have

20 + 30x = 36 + 28x

**Step 4 : **

To get rid of "28x" on the right side, we have to subtract "28x" from both sides.

aaaaaaaaaaaaaaaaaa20 + 30x = 36 + 28x aaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaa -28x -28x aaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa------------------------aaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaa 20 + 2x = 36 aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa------------------------

**Step 5 : **

To get rid of "20" on the left side, subtract 20 from both sides.

aaaaaaaaaaaaaaaaaaaaa20 + 2x = 36 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa -20 -20 aaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaa----------------aaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa 2x = 16 aaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa ----------------

**Step 6 : **

To get rid of "2" multiplied by x, divide by 2 on both sides.

2x / 2 = 16 / 2

x = 8

Hence, the total cost charged by both of them is same for 8 days.

After having gone through the stuff given above, we hope that the students would have understood "Solving an equation with variables on both sides".

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