Question

What is the solution to $\dfrac{2}{3}x - 2 = - 5x + 1$?

Answer 1

Hint: To solve this problem isolate all terms containing $x$ on one side of the equation and all other terms on the other side of the equation get the terms over a common denominator and then solved for $x$ while keeping the equation balanced. Solve the equations. And easily we get the solution for the given problem.

Complete step-by-step solution:
To solve this problem isolate all terms containing $x$ on the side of the equation.
$\dfrac{2}{3}x - 2 = - 5x +1$
$\dfrac{2}{3}x + 5x = 1 + 2$
Solve the equation
$\dfrac{2}{3}x + 5x = 3$
Get the $x$terms over a common denominator and then solve for $x$ while keeping the equation balanced.
$\dfrac{2}{3}x + 5x = 3$
Multiple terms $5x$ by $\dfrac{3}{3}$ to equalize the equation we get,
However we need to multiply the top and bottom by some number which is multiplication by $1$.
Therefore,
$\dfrac{2}{3}x + \left( {\dfrac{3}{3}} \right)5x = 3$
$\Rightarrow \dfrac{2}{3}x + \dfrac{{15x}}{3} = 3$
$\Rightarrow \dfrac{{17x}}{3} = 3$
Multiply both sides by$\left( {\dfrac{3}{{17}}} \right)$, we have
$\left( {\dfrac{3}{{17}}} \right)\dfrac{{17}}{3}x = 3.\left( {\dfrac{3}{{17}}} \right)$
$\Rightarrow \dfrac{{17x}}{{17}} = \dfrac{9}{{17}} $
$ \Rightarrow x = \dfrac{9}{{17}} $
Therefore the solution for the given problem is $x = \dfrac{9}{{17}}$
Additional information:
In this additional information we are going to learn to solve the problem in another simple method.
Combine multiplied terms into a single fraction.
$\dfrac{2}{3}x - 2 = - 5x + 1$
Add the number two in both side of the equation one we get,
$\dfrac{2}{3}x - 2 = - 5x + 1$
$\dfrac{2}{3}x - 2 + 2 = - 5x + 1 + 2$
To simplify the equation add the numbers we get,
$\dfrac{{2x}}{3} = - 5x + 3$
Add the $5x$ to both sides of the equation we get,
$\dfrac{{2x}}{3} + 5x = - 5x + 5x + 3$
Remove the same value which has different sign we have,
$\dfrac{{2x}}{3} + 5x = 3$
Multiply all terms by the same value to eliminate fraction denominators.
$ \dfrac{{2x}}{3} + 5x = 3 \\
  3\left( {\dfrac{{2x}}{3} + 5x} \right) = 3 \times 3 \\ $
Cancel the multiplied terms that are in the denominator we have,
\[3.\dfrac{{2x}}{3} + 3\times 5x = 3\times 3\]
$2x + 3\times 5x = 3\times 3$
Multiply the numbers
$2x + 3\times 5x = 3\times 3$
$2x + 15x = 3\times 3$
Combine like terms we have,
$17x = 9$
Divide both sides of the equation by the some term we have
$
  \dfrac{{17x}}{{17}} = \dfrac{9}{{17}} \\
  x = \dfrac{9}{{17}} \\
 $
Therefore the solution for the given problem is
$x = \dfrac{9}{{17}}$

Note: In mathematics, to solve an equation to find its solutions, which are the values that fulfill the condition stated by the equation, consitting generally of two expressions are related by an equals sign. When we search for a solution one or more variables are given as unknown values. To solve the equations we have a cancellation method, substitution method and cross multiplication method.