1Cademy - Solving $$m = \frac{x-2}{y-3}$$ for $$y$$

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Strategy to Solve Equations with Rational Expressions

Example

Solving $m = \frac{x-2}{y-3}$ for $y$

Solve the rational equation $m = \frac{x-2}{y-3}$ for the variable $y$ by applying the strategy for solving equations with rational expressions. This formula is based on the slope formula, where $m$ represents the slope of a line passing through the points $(2, 3)$ and $(x, y)$ .

Step 1 — Identify restricted values. The variable $y$ appears in the denominator $(y - 3)$ , so $y \neq 3$ .

Step 2 — Clear the fractions. The LCD is $(y - 3)$ . Multiply both sides by $(y - 3)$ :

$(y - 3) \cdot m = (y - 3) \cdot \frac{x - 2}{y - 3}$

Step 3 — Simplify. On the right, the $(y - 3)$ factors cancel:

$m(y - 3) = x - 2$

Distribute $m$ on the left:

$my - 3m = x - 2$

Step 4 — Isolate the term with $y$ . Add $3m$ to both sides:

$my = x - 2 + 3m$

Step 5 — Divide both sides by $m$ to isolate $y$ :

$y = \frac{x - 2 + 3m}{m}$

This example is more involved than the previous one because the target variable $y$ appears inside a binomial denominator $(y - 3)$ . After clearing the fraction by multiplying both sides by the LCD, distributing is required to separate the $y$ -term from the constant. Collecting the $y$ -term on one side and then dividing by its coefficient $m$ produces the final result.

Updated 2026-04-21

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OpenStax Elementary Algebra

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