Question

Solve the equation \[7 - \dfrac{2}{3}x = 4\dfrac{8}{3} - 5x\].

Answer 1

Hint: To solve the equation of the form $ax = b$, we are required to divide both sides of the equation by $a$ to get the equation of the form $x = \dfrac{b}{a}$. While rearranging an equation, proper algebraic rules should be followed, that is, an appropriate inverse operation is used.

Complete step-by-step solution:
The given equation is \[7 - \dfrac{2}{3}x = 4\dfrac{8}{3} - 5x\].
We are required to solve the above equation using systematic steps.
First convert the mixed fractions into improper fraction,
\[7 - \dfrac{2}{3}x = \dfrac{{4 \times 3 + 8}}{3} - 5x\]
\[\Rightarrow 7 - \dfrac{2}{3}x = \dfrac{{20}}{3} - 5x\]
Rearrange the above equation by taking all the terms consisting of $x$ on one side of the equality and all the remaining terms on the other side of the equality by changing the sign properly.
Take the term $ - 5x$ from the right side of the to the left side of the equation by converting it into $5x$.
\[7 - \dfrac{2}{3}x + 5x = \dfrac{{20}}{3}\]
Now, take the term $7$ from the left side of the equation to the right side of the equation by converting it into $ - 7$.
 \[ - \dfrac{2}{3}x + 5x = \dfrac{{20}}{3} - 7\]
Take $x$ common from the left side of the equation,
 \[\left( {5 - \dfrac{2}{3}} \right)x = \dfrac{{20}}{3} - 7\]
Solve the fraction terms on both sides of the equation,
\[\Rightarrow \left( {\dfrac{{15 - 2}}{3}} \right)x = \dfrac{{20 - 21}}{3}\]
\[\Rightarrow \left( {\dfrac{{13}}{3}} \right)x = \dfrac{{ - 1}}{3}\]
We notice that this equation is of the form $ax = b$, so we are required to divide both sides of the equation by $a$ to get the equation of the form $x = \dfrac{b}{a}$.
In the above equation we get, $a = \dfrac{{13}}{3}$
First, divide both sides of the equation by \[\dfrac{{13}}{3}\], we get
\[x = \left( {\dfrac{{ - 1}}{3}} \right) \div \left( {\dfrac{{13}}{3}} \right)\]
To solve the above division, multiply the first term by the reciprocal of the second term.
\[x = \left( {\dfrac{{ - 1}}{3}} \right) \times \left( {\dfrac{3}{{13}}} \right)\]
Simplify the above equation by cancelling $3$from the numerator and denominator on the left side of the equation.
\[x = - \dfrac{1}{{13}}\]

Note: When rearranging the terms in an equation, using the inverse operation rule, the addition term on one side of the equality changes to the negative term on the other side of the equality. Similarly, like this, negative changes to positive, division changes to multiplication and multiplication changes to division.