Question

solve the following linear equations.
i) \[\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}\]
ii) \[\dfrac{n}{2} - \dfrac{{3n}}{4} + \dfrac{{5n}}{6} = 21\]
iii) \[x + 7 - \dfrac{{8x}}{3} = \dfrac{{17}}{6} - \dfrac{{5x}}{2}\]

Answer 1

Hint: A linear equation is a mathematical equation that consists of numbers and variables with a relationship of multiplication, addition, subtraction and division between them. A linear equation consists of one as their highest power. So, to solve any linear equation, we have to arrange all the variables on one side and all the numbers on another side. For example \[x + 3 = 0\] is a linear equation, where it consists of variable x and number 3 and equated to zero.


Complete step by step solution
i)
Given:
The linear equation is \[\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}\]. On simplifying the equation, we get,
\[\begin{array}{c}
\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}\\
\dfrac{x}{2} - \dfrac{x}{3} = \dfrac{1}{4} + \dfrac{1}{5}\\
\dfrac{{3x - 2x}}{6} = \dfrac{{5 + 4}}{{20}}
\end{array}\]
On further solving the above expression, we get,
\[\begin{array}{c}
\dfrac{x}{6} = \dfrac{9}{{20}}\\
x = \dfrac{{54}}{{20}}\\
x = 2.7
\end{array}\]
Therefore, the value of the x is 2.7.
ii)
Given:
The linear equation is \[\dfrac{n}{2} - \dfrac{{3n}}{4} + \dfrac{{5n}}{6} = 21\]. On simplifying the equation, we get,
\[\begin{array}{c}
\dfrac{n}{2} - \dfrac{{3n}}{4} + \dfrac{{5n}}{6} = 21\\
\dfrac{{6n - 9n + 10n}}{{12}} = 21\\
\dfrac{{7n}}{{12}} = 21
\end{array}\]
On further solving the above expression, we get,
\[\begin{array}{c}
n = \dfrac{{21 \times 12}}{7}\\
n = 36
\end{array}\]
Therefore, the value of the n is 36.
iii)
Given:
The linear equation is \[x + 7 - \dfrac{{8x}}{3} = \dfrac{{17}}{6} - \dfrac{{5x}}{2}\]. On simplifying the equation, we get,
\[\begin{array}{c}
x + 7 - \dfrac{{8x}}{3} = \dfrac{{17}}{6} - \dfrac{{5x}}{2}\\
x - \dfrac{{8x}}{3} + \dfrac{{5x}}{2} = \dfrac{{17}}{6} - 7\\
\dfrac{{6x - 16x + 15x}}{6} = \dfrac{{17 - 42}}{6}
\end{array}\]
On further solving the above expression, we get,
\[\begin{array}{c}
\dfrac{{5x}}{6} = \dfrac{{ - 25}}{6}\\
x = \dfrac{{ - 25}}{5}\\
x = - 5
\end{array}\]
Therefore, the value of the x is \[ - 5\].


Note: In this solution, we should be careful while solving because the equation consists of positive and negative symbols so while bringing the right side values to left side and left side values to right sides it is important to change their signs also. In the sub part (iii), we can see that 17 is subtracted from the 42 so at this point we have to add a minus symbol before the 25 because 17 is a smaller number than 42 and similarly while subtracting 6x and 8x, we will get -2x but after adding with 15x we will get 13x which is a positive value because here already a minus symbol is allocated to smaller number.