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put $xy = ax + by$ in form of $y=mx+c$ where $Y = \frac{x}{y}$ and $X = x$

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Find a value using two-variable equations

Full solution

Q. put

xy = ax + by

in form of

y=mx+c

where

Y = \frac{x}{y}

and

X = x

Rewrite equation: We are given the equation $xy = ax + by$ and we need to solve for $y$ in terms of $x$ to get it in the form $y = mx + c$ . $\newline$ First, let's rewrite the equation to isolate terms with $y$ on one side. $\newline$ $xy - by = ax$
Factor out $y$ : Now, factor out $y$ from the left side of the equation. $y(x - b) = ax$
Divide by $(x - b)$ : Next, divide both sides of the equation by $(x - b)$ to solve for $y$ . $y = \frac{ax}{(x - b)}$
Express $y$ in terms: Now, we need to express $y$ in terms of $X$ and $Y$ , where $Y = \frac{x}{y}$ and $X = x$ . First, let's solve the equation $Y = \frac{x}{y}$ for $y$ . $y = \frac{x}{Y}$
Substitute Y into equation: Substitute $Y = \frac{x}{y}$ into the equation $y = \frac{ax}{(x - b)}$ . $\newline$ $y = \frac{ax}{(X - b)}$
Replace $X$ with $x$ : Since we have $y$ in terms of $X$ , we can now replace $X$ with $x$ to get the final form. $\newline$ $y = \frac{ax}{x - b}$ $\newline$ This is the equation in the form $y = mx + c$ , where $m = \frac{a}{x - b}$ and $c = 0$ .

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