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# Find the value of x in the equation, $\dfrac{2}{3}x + 4 = \dfrac{3}{5}x - 2$

Last updated date: 11th Aug 2024
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Hint: There are various ways this question can be solved for x. One way is to take LCM on both the sides of the equation. Once LCM is done, you’ll have 2 fractions on both sides. Cross multiply and you’ll see a linear equation in place. Last step will be the use of a basic operation that is to be applied on both the sides to get the final answer. In case of more complex equations, one might have to use a combination of different operations, i.e., multiplication, division, addition and subtraction.

Let us denote the sides as Left-hand side (LHS) and Right-hand side (RHS).
$\dfrac{2}{3}x + 4 = \dfrac{3}{5}x - 2$
Now, after taking LCM on the LHS and the RHS the resulting equation will be,
$\Rightarrow \dfrac{{2x + 12}}{3} = \dfrac{{3x - 10}}{5}$
Cross multiplying on both sides,
$\Rightarrow 5 \times (2x + 12) = 3 \times (3x - 10) \\ \Rightarrow 10x + 60 = 9x - 30 \;$
Now, take the variable on the LHS and the constants on the RHS.
$\Rightarrow x = - 90$
Thus, the value of $x = - 90$ .
Alternate Method: One can also solve this by taking the fractions on the left-hand side and the constants on the right.
$\dfrac{2}{3}x + 4 = \dfrac{3}{5}x - 2 \\ \Rightarrow \dfrac{2}{3}x - \dfrac{3}{5}x = - 2 - 4 \;$
Now, taking the LCM of the denominators of the fractions and making changes in the numerators accordingly, the equation changes to
$\Rightarrow \dfrac{{10x - 9x}}{{15}} = - 6$
Cross multiplying, we have
$\Rightarrow x = - 90$
So, the correct answer is “ x = - 90”.

Note: A linear equation is an equation that can be put in the form of the variables and the coefficients, mostly real numbers. Coefficients may be equation parameters or arbitrary expressions, provided that they do not include any of the variables. First order equations are linear equations.