SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES WORKSHEET

Question 1 :

Solve for t : 

5t - 9  =  -3t + 7

Question 2 :

Solve for x : 

3x - 1  =  x + 5

Question 3 :

Solve for n : 

3n/4 + 16  =  2 - n/8

Question 4 :

Solve for n : 

4(2a - 1) = -10(a - 5)

Question 5 :

Solve for x : 

-7x - 3x + 2  =  -8x - 8

Question 6 :

Solve for k :

5(k - 3) - 7(6 - k)  =  24 - 3(8 - k) - 3

Question 7 : 

Solve for x : 

(x + 15)(x - 3) - (x² - 6x + 9)  =  30 - 15(x - 1)

Question 8 :

Solve for x :  

4x/5 - 7/4  =  x/5 + x/4

Question 9 :

Solve for x :  

(x - 2)/2 + (x + 10)/9  =  5

Question 10 : 

Solve the following equation : 

(1/2)(8y - 6)  =  5y - (y + 3)

Question 11 : 

Solve the following equation : 

2(1 - x) + 5x  =  3(x + 1)

Question 12 :

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?

Answers

1. Answer :

5t - 9  =  -3t + 7

Add 3t to each side. 

8t - 9  =  7

Add 9 to each side.

8t  =  16

Divide each side by 8. 

t  =  2

2. Answer :

3x - 1  =  x + 5

Subtract x from each side. 

2x - 1  =  5

Add 1 to each side. 

2x  =  6

Divide each side by 2.

x  =  3

3. Answer :

3n/4 + 16  =  2 - n/8

Least common multiple of the denominators (4 and 8) is 8.

Multiply each side by 8 to get rid of the denominators 4 and 8.

8[3n/4 + 16]  =  8[2 - n/8]

Use the distributive property. 

8(3n/4) + 8(16)  =  8(2) - 8(n/8)

6n + 128  =  16 - n

Add n to each side. 

7n + 128  =  16

Subtract 128 from each side.

7n  =  -112

Divide each side by 7.

n  =  -16

4. Answer :

4(2a - 1) = -10(a - 5)

Use the distributive property. 

4(2a) - 4(1)  =  -10(a) - 10(-5)

8a - 4  =  -10a + 50

Add 10a to each side. 

18a - 4  =  50

Add 4 to each side. 

18a  =  54

Divide each side by 18.

a  =  3

5. Answer :

-7x - 3x + 2  =  -8x - 8

Simplify.

-10x + 2  =  -8x - 8

Add 10x to each side. 

2  =  2x - 8

Add 8 to each side.

10  =  2x

Divide each side by 2.

5  =  x

6. Answer :

5(k - 3) - 7(6 - k)  =  24 - 3(8 - k) - 3

Use distributive property. 

5k - 15 - 42 + 7k  =  24 - 24 + 3k - 3

Simplify. 

12k - 57  =  3k - 3

Subtract 3k from each side. 

9k - 57  =  -3

Add 57 to each side.

9k  =  54

Divide each side by 9.

k  =  6

7. Answer :

(x + 15)(x - 3) - (x² - 6x + 9)  =  30 - 15(x - 1)

Simplify. 

x² + 12x - 45 - x² + 6x - 9  =  30 - 15x + 15

18x - 54  =  -15x + 45

Add 15x to each side. 

33x - 54  =  45

Add 54 to each side.

33x  =  99

Divide each side by 33.

x  =  3

8. Answer :

4x/5 - 7/4  =  x/5 + x/4

The least common multiple of the denominators (4 and 5) is 20.

Multiply each side by 20 to get rid of the denominators 4 and 8.

20[4x/5 - 7/4]  =  20[x/5 + x/4]

20(4x/5) - 20(7/4)  =  20(x/5) + 20(x/4)

16x - 35  =  4x + 5x

16x - 35  =  9x

Subtract 9x from each side. 

7x - 35  =  0

Add 35 to each side.

7x  =  35

Divide each side by 7.

x  =  5

9. Answer :

(x - 2)/2 + (x + 10)/9  =  5

The least common multiple of the denominators (2 and 9) is 18.

Multiply each side 18 to get rid of the denominators 2 and 9. 

18[(x - 2)/2 + (x + 10)/9]  =  18(5)

18(x - 2)/2 + 18(x + 10)/9  =  90

9(x - 2) + 2(x + 10)  =  90

9x - 18 + 2x + 20  =  90

11x + 2  =  90

Subtract 2 from each side. 

11x  =  88

Divide each side by 11. 

x  =  8

10. Answer :

(1/2)(8y - 6)  =  5y - (y + 3)

Simplify both sides. 

4y - 3  =  5y - y - 3

4y - 3  =  4y - 3  

Subtract 4y from each side. 

-3  =  -3

The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions. 

11. Answer :

2(1 - x) + 5x  =  3(x + 1)

Simplify both sides. 

2 - 2x + 5x  =  3x + 3

2 + 3x  =  3x + 3

Subtract 3x from each side. 

2  =  3

The above result is false. Because 2 is not equal to 3. Because the result we get at the last step is false, the given equation has no solution.  

12. Answer :

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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