Answer:
x = -10
Step-by-step explanation:
Solve for x:
(2 (6 x + 9))/3 + x = (2 (5 x - 10))/5 + 2 x
Put each term in (2 (6 x + 9))/3 + x over the common denominator 3: (2 (6 x + 9))/3 + x = (2 (6 x + 9))/3 + (3 x)/3:
(2 (6 x + 9))/3 + (3 x)/3 = (2 (5 x - 10))/5 + 2 x
(2 (6 x + 9))/3 + (3 x)/3 = (2 (6 x + 9) + 3 x)/3:
(2 (6 x + 9) + 3 x)/3 = (2 (5 x - 10))/5 + 2 x
2 (6 x + 9) = 12 x + 18:
(12 x + 18 + 3 x)/3 = (2 (5 x - 10))/5 + 2 x
Grouping like terms, 12 x + 3 x + 18 = (12 x + 3 x) + 18:
((12 x + 3 x) + 18)/3 = (2 (5 x - 10))/5 + 2 x
12 x + 3 x = 15 x:
(15 x + 18)/3 = (2 (5 x - 10))/5 + 2 x
Put each term in (2 (5 x - 10))/5 + 2 x over the common denominator 5: (2 (5 x - 10))/5 + 2 x = (2 (5 x - 10))/5 + (10 x)/5:
(15 x + 18)/3 = (2 (5 x - 10))/5 + (10 x)/5
(2 (5 x - 10))/5 + (10 x)/5 = (2 (5 x - 10) + 10 x)/5:
(15 x + 18)/3 = (2 (5 x - 10) + 10 x)/5
2 (5 x - 10) = 10 x - 20:
(15 x + 18)/3 = (10 x - 20 + 10 x)/5
10 x + 10 x = 20 x:
(15 x + 18)/3 = (20 x - 20)/5
Multiply both sides by 15:
(15 (15 x + 18))/3 = (15 (20 x - 20))/5
15/3 = (3×5)/3 = 5:
5 (15 x + 18) = (15 (20 x - 20))/5
15/5 = (5×3)/5 = 3:
5 (15 x + 18) = 3 (20 x - 20)
Expand out terms of the left hand side:
75 x + 90 = 3 (20 x - 20)
Expand out terms of the right hand side:
75 x + 90 = 60 x - 60
Subtract 60 x from both sides:
(75 x - 60 x) + 90 = (60 x - 60 x) - 60
75 x - 60 x = 15 x:
15 x + 90 = (60 x - 60 x) - 60
60 x - 60 x = 0:
15 x + 90 = -60
Subtract 90 from both sides:
15 x + (90 - 90) = -90 - 60
90 - 90 = 0:
15 x = -90 - 60
-90 - 60 = -150:
15 x = -150
Divide both sides of 15 x = -150 by 15:
(15 x)/15 = (-150)/15
15/15 = 1:
x = (-150)/15
The gcd of -150 and 15 is 15, so (-150)/15 = (15 (-10))/(15×1) = 15/15×-10 = -10:
Answer: x = -10
