Full solution

Q. Solve for

x

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\frac{2 x}{5}+\frac{1}{6}=\frac{3 x+1}{2}

Find Common Denominator: We have the equation $(2x)/(5) + (1)/(6) = (3x+1)/(2)$ . The first step is to find a common denominator for the fractions on the left side of the equation to combine them. The least common multiple of $5$ and $6$ is $30$ .
Convert Fractions to LCD: Convert the fractions $(\frac{2x}{5})$ and $(\frac{1}{6})$ to have the common denominator of $30$ . This gives us $(\frac{2x \times 6}{5 \times 6}) + (\frac{1 \times 5}{6 \times 5}) = (\frac{3x+1}{2})$ .
Simplify Fractions: Simplify the fractions to get $(\frac{12x}{30}) + (\frac{5}{30}) = (\frac{3x+1}{2})$ .
Combine Fractions: Combine the fractions on the left side of the equation to get a single fraction. This gives us $(12x + 5)/(30) = (3x+1)/(2)$ .
Cross-Multiply: To solve for $x$ , we can cross-multiply. This means multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal to each other. So, $(12x + 5) \times 2 = (3x + 1) \times 30$ .
Perform Multiplication: Perform the multiplication to get $24x + 10 = 90x + 30$ .
Isolate $x$ : To isolate $x$ , we need to get all the $x$ terms on one side and the constants on the other. Subtract $24x$ from both sides to get $10 = 66x + 30$ .
Subtract Constants: Subtract $30$ from both sides to get $-20 = 66x$ .
Divide by $66$ : Divide both sides by $66$ to solve for $x$ . This gives us $x = -\frac{20}{66}$ .
Simplify Fraction: Simplify the fraction $-\frac{20}{66}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ . This gives us $x = -\frac{10}{33}$ .