SOLVING AN EQUATION WITH VARIABLES ON BOTH SIDES

Example 1 :

Solve for a :

-3 + 5a = 4a - 10

Solution :

-3 + 5a = 4a - 10

Subtract 4a from both sides.

-3 + a = -10

Add 3 to both sides.

a = -7

Example 2 :

Solve for x :

3x - 1 = x + 5

Solution :

3x - 1 = x + 5

Subtract x from both sides.

2x - 1 = 5

Add 1 to both sides.

2x = 6

Divide both sides by 2.

x = 3

Example 3 :

Solve for y :

1 - 3y = 3y + 1

Solution :

1 - 3y = 3y + 1

Subtract 3y from both sides.

1 - 6y = 1

Subtract 1 from both sides.

6y = 0

Divide both sides by 2.

y = 0

Example 4 :

Solve for n :

4n - 1 = 6n + 8 - 8n + 15

Solution :

4n - 1 = 6n + 8 - 8n + 15

Simplify the expression on the right side.

4n - 1 = -2n + 23

Add 2n from both sides.

6n - 1 = 23

Add 1 to both sides.

6n = 24

Divide both sides by 6.

n = 4

Example 5 :

Solve for x :

1 - x = 16 - 4x

Solution :

1 - x = 16 - 4x

Add 4x to both sides.

1 - x = 16 - 4x

1 + 3x = 16

Subtract 1 from both sides.

3x = 15

Divide both sides by 3.

x = 5

Example 6 :

Solve for z :

-40 + 2z = 4z - 8(z + 8)

Solution :

-40 + 2z = 4z - 8(z + 8)

Use the distributive property on the right side.

-40 + 2z = 4z - 8z - 64

Simplify the expression on the right side.

-40 + 2z = -4z - 64

Add 4z to both sides.

-40 + 6z = -64

Add 40 to both sides.

6z = -24

Divide both sides by 6.

z = -4

Example 7 :

Solve for x :

7x - 8(-x + 7) = -16 + 7x

Solution :

7x - 8(-x + 7) = -16 + 7x

Use the distributive property on the left side.

7x + 8x - 56 = -16 + 7x

Simplify the expression on the left side.

15x - 56 = -16 + 7x

Subtract 7x from both sides.

8x - 56 = -16

Add 56 to both sides.

8x = 40

Divide both sides by 8.

x = 5

Example 8 :

Solve for r :

-6(2r - 2) = -8r + 40

Solution :

-6(2r - 2) = -8r + 40

Use the distributive property on the left side.

-12r + 12 = -8r + 40

Add 8r to both sides.

-4r + 12 = 40

Subtract 12 from both sides.

-4r = 28

Divide both sides by -4.

r = -7

Example 9 :

Solve for x :

2(1 + 6x) + 2(3 - 6x) = 7x - 5x

Solution :

2(1 + 6x) + 2(3 - 6x) = 7x - 5x

Use the distributive property on the left side.

2 + 12x + 6 - 12x = 7x - 5x

Simplify the expressions on both sides.

8 = 2x

Divide both sides by 2.

4 = x

Example 10 :

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 : 

Solve for x : 

20 + 30x = 36 + 28x

Subtract 28x from each side. 

20 + 2x = 36

Subtract 20 from each side. 

2x = 16

Divide each side by 2. 

x = 8

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

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