Question - Solving System of Equations for y
Solution:
To solve the given system of equations for $$ y $$, follow these steps:The system of equations given are:\[ y = \frac{4}{7}x \]\[ \frac{2}{3}x = y + \frac{5}{7} \]First, we can use the first equation to substitute $$ x $$ in terms of $$ y $$ in the second equation. To find $$ x $$ in terms of $$ y $$ from the first equation, we rearrange the equation as:\[ y = \frac{4}{7}x \]\[ x = \frac{7}{4}y \]Now we substitute $$ x $$ into the second equation:\[ \frac{2}{3}\left(\frac{7}{4}y\right) = y + \frac{5}{7} \]To solve for $$ y $$, we now multiply the terms:\[ \frac{14}{12}y = y + \frac{5}{7} \]Simplify the left side by reducing the fraction:\[ \frac{7}{6}y = y + \frac{5}{7} \]To solve for y, get all the terms involving y on one side of the equation:\[ \frac{7}{6}y - y = \frac{5}{7} \]Since $$ y $$ is the same as $$ \frac{6}{6}y $$, we can rewrite the equation as:\[ \frac{7}{6}y - \frac{6}{6}y = \frac{5}{7} \]Combine the y terms:\[ \frac{1}{6}y = \frac{5}{7} \]Finally, solve for $$ y $$ by multiplying both sides by the reciprocal of $$ \frac{1}{6} $$, which is 6:\[ y = \frac{5}{7} \times 6 \]\[ y = \frac{5 \times 6}{7} \]\[ y = \frac{30}{7} \]So the value of $$ y $$ is $$ \frac{30}{7} $$.The correct answer is 30/7.
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