Question

Solve the following equation
\[6(3x + 2) - 5(6x - 1) = 3(x - 8) - 5(7x - 6) + 9x\]

Answer 1

Hint: In the given problem we need to solve this for ‘x’. We can solve this using the transposition method. We first expand the brackets in both sides of the equation and then we add or subtract the like terms in LHS and RHS. Then we group the ‘x’ terms on one side and constants on the other side of the equation. After simplifying we will have the desired result.

Complete step-by-step solution:
Given, \[6(3x + 2) - 5(6x - 1) = 3(x - 8) - 5(7x - 6) + 9x\].
Now expanding the brackets we have,
\[18x + 12 - 30x + 5 = 3x - 24 - 35x + 30 + 9x\]
Now adding the like terms and constant on both sides of the equation,
\[17 - 12x = - 23x + 6\]
We transpose ‘17’ which is present on the left-hand side of the equation to the right-hand side of the equation by subtracting ‘17’ on the right-hand side of the equation.
\[ - 12x = - 23x + 6 - 17\]
Similarly, we transpose or shift the term $-23x$ to the left side of the equation by adding $23x$ in the left-hand side.
\[ - 12x + 23x = 6 - 17\]
\[11x = - 11\]
Now dividing the whole equation by 11
\[ \Rightarrow x = - 1\]. This is the required answer.

Note: We can cross check our obtained answer. To cross check the answer we substitute the obtained answer in the given problem, if we obtained LHS is equal to RHS then our obtained answer is correct.
\[6(3x + 2) - 5(6x - 1) = 3(x - 8) - 5(7x - 6) + 9x\]
\[\Rightarrow 6(3( - 1) + 2) - 5(6( - 1) - 1) = 3(( - 1) - 8) - 5(7( - 1) - 6) + 9( - 1)\]
\[\Rightarrow 6( - 3 + 2) - 5( - 6 - 1) = 3( - 1 - 8) - 5( - 7 - 6) - 9\]
\[\Rightarrow 6( - 1) - 5( - 7) = 3( - 9) - 5( - 13) - 9\]
\[\Rightarrow - 6 + 35 = - 27 + 65 - 9\]
Simplifying we have,
\[ \Rightarrow 29 = 129\].
That is LHS=RHS. Hence the obtained is correct.
We need to be careful in simplification. While multiplying a negative number with the negative numbers we will get a positive number. Also when we multiply a negative (positive) number with a positive (negative) number we will have a negative number.
In the above, we did the transpose of addition and subtraction. Similarly, if we have multiplication we use division to transpose. If we have division, we use multiplication to transpose. Follow the same procedure for these kinds of problems.