Question - Solving for Variables in Similar Triangles with Proportions
Solution:
The image shows two triangles with corresponding sides labeled with algebraic expressions. To solve for the values of $$ x $$ and $$ y $$, we must assume that these triangles are similar based on the way they're labeled (for instance, the triangles could be indicated as similar by a given problem statement or corresponding angles being marked as congruent). Similar triangles have corresponding sides that are proportional. Let's set up the proportion based on the sides provided:$$\frac{x}{3y} = \frac{x + 3}{y + 1}$$Now we cross-multiply to solve for $$ y $$:$$x(y + 1) = 3y(x + 3)$$Expand both sides:$$xy + x = 3xy + 9y$$Now, solve for $$ y $$ in terms of $$ x $$:$$xy - 3xy = 9y - x$$$$-2xy = 9y - x$$Now, since we want $$ y $$ in terms of $$ x $$, we need to factor $$ y $$ out on the left-hand side:$$y(-2x - 9) = -x$$Divide both sides by $$-2x - 9$$ to isolate $$ y $$:$$y = \frac{-x}{-2x - 9}$$This gives us $$ y $$ in terms of $$ x $$. However, you may have additional information or a system of equations to solve for the specific values of $$ x $$ and $$ y $$, which isn't provided in the image. If that's the case, use that additional information to find the exact values of $$ x $$ and $$ y $$. If the image does not provide enough information to solve the system definitively and you require further context or other equations, please provide them to proceed.
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