Question

How to solve the following equations?
$$1)5 + x = \dfrac{1}{2} - \dfrac{1}{3}$$,
 $$2)\dfrac{1}{4} + x = \dfrac{1}{2} - \dfrac{1}{3}$$,
$$3)1 - \dfrac{1}{2} + m = \dfrac{3}{4} + \dfrac{1}{2}$$.

Answer 1

Hint: Here in this question, we have to solve the given equation and it is in the form of an algebraic equation having a variable $$x$$ or $$m$$. Solving this equation, we have to find the unknown value $$x$$ or $$m$$ by using the basic arithmetic operation like multiplication and division to find the required value.

Complete step by step answer:
The given equation is an algebraic equation. The algebraic equation is a combination of variable and constant and which has an equal sign. So, we use multiplication and division or arithmetic operations and solve for further
Now consider the given equations one by one
 $$1)5 + x = \dfrac{1}{2} - \dfrac{1}{3}$$
Take LCM 6 on RHS
$$ \Rightarrow \,\,5 + x = \dfrac{{3 - 2}}{6}$$
$$$$
The value of $$\dfrac{1}{6} = 0.167$$, then we have
$$ \Rightarrow \,\,5 + x = 0.167$$
Subtract both sides by $$5$$.
$$ \Rightarrow \,\,x = 0.167 - 5$$
On simplification, we get
$$\therefore \,\,\,\,x = - 4.833$$
Therefore, the value of $$5 + x = \dfrac{1}{2} - \dfrac{1}{3}$$ is $$ - 4.833$$.

$$2)\dfrac{1}{4} + x = \dfrac{1}{2} - \dfrac{1}{3}$$
Or
$$\dfrac{1}{4} + x = \dfrac{1}{2} - \dfrac{1}{3}$$
On simplification, we have
$$ \Rightarrow \,\,0.25 + x = \dfrac{1}{2} - \dfrac{1}{3}$$
 Take LCM 6 on RHS
$$ \Rightarrow \,\,0.25 + x = \dfrac{{3 - 2}}{6}$$
$$ \Rightarrow \,\,0.25 + x = \dfrac{1}{6}$$
The value of $$\dfrac{1}{6} = 0.167$$, then we have
$$ \Rightarrow \,\,0.25 + x = 0.167$$
Subtract both sides by $$0.25$$.
$$ \Rightarrow \,\,x = 0.167 - 0.25$$
On simplification, we get
$$\therefore \,\,\,\,x = - 0.0833$$
Therefore, the value of $$\dfrac{1}{4} + x = \dfrac{1}{2} - \dfrac{1}{3}$$ is $$ - 0.083$$.
$$3)1 - \dfrac{1}{2} + m = \dfrac{3}{4} + \dfrac{1}{2}$$
Or
$$1 - 0.5 + m = \dfrac{3}{4} + \dfrac{1}{2}$$
On simplification, we have
$$ \Rightarrow \,\,0.5 + m = \dfrac{3}{4} + \dfrac{1}{2}$$
 Take LCM 4 on RHS
$$ \Rightarrow \,\,0.5 + m = \dfrac{{3 + 2}}{4}$$
$$ \Rightarrow \,\,0.5 + m = \dfrac{5}{4}$$
The value of $$\dfrac{5}{4} = 1.25$$, then we have
$$ \Rightarrow \,\,0.5 + m = 1.25$$
Subtract both sides by $$0.5$$.
$$ \Rightarrow \,\,m = 1.25 - 0.5$$
On simplification, we get
$$\therefore \,\,\,\,m = 0.75$$
Therefore, the value of $$1 - \dfrac{1}{2} + m = \dfrac{3}{4} + \dfrac{1}{2}$$ is $$ 0.75$$.

Note:
If the algebraic expression contains only one unknown, we determine the value by using simple multiplication and division.
There are three methods to solve linear equations in one variable:
1. Trial and error
2. Transposition method
3. Inverse operation