Why multiply the equation $1 - 2/3 y = 7/3$ by 3?

Conditions

• The equation contains fractional terms with denominators 1 and 3.
• All terms on both sides of the equation are multiplied by the same nonzero number (3).

Reasoning, step by step

1. Identify the denominators in the equation $1 - \frac{2}{3}y = \frac{7}{3}$, which are 1 and 3.
2. Determine the common denominator, which is 3.
3. Multiply every term in the equation by 3: $3(1) - 3(\frac{2}{3}y) = 3(\frac{7}{3})$.
4. Simplify the products to obtain the fraction-free equation: $3 - 2y = 7$.
5. Solve the resulting one-variable linear equation for $y$.

Example

Multiplying $1 - \frac{2}{3}y = \frac{7}{3}$ by 3 yields $3 - 2y = 7$. Subtracting 3 from both sides gives $-2y = 4$, and dividing by $-2$ gives $y = -2$.

Common misconceptions

• Forgetting to multiply the constant term (1) by 3, which would incorrectly leave it as 1 instead of 3.
• Assuming that multiplying by the common denominator changes the value of the equation rather than producing an equivalent equation.

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