c) ) (3)/(x)+(5)/(y)=1 (4)/(x)+(3)/(y)=-17

To solve the system of equations:<br /><br />\[<br />\begin{cases}<br />\frac{3}{x} + \frac{5}{y} = 1 \\<br />\frac{4}{x} + \frac{3}{y} = -17<br />\end{cases}<br />\]<br /><br />we can use the method of substitution or elimination. Let's use the elimination method.<br /><br />First, let's multiply the first equation by 3 and the second equation by 5 to make the coefficients of \(\frac{3}{y}\) and \(\frac{4}{y}\) equal:<br /><br />\[<br />\begin{cases}<br />3 \left( \frac{3}{x} + \frac{5}{y} \right) = 3 \cdot 1 \\<br />5 \left( \frac{4}{x} + \frac{3}{y} \right) = 5 \cdot (-17)<br />\end{cases}<br />\]<br /><br />This simplifies to:<br /><br />\[<br />\begin{cases}<br />\frac{9}{x} + \frac{15}{y} = 3 \\<br />\frac{20}{x} + \frac{15}{y} = -85<br />\end{cases}<br />\]<br /><br />Next, subtract the first equation from the second equation to eliminate \(\frac{15}{y}\):<br /><br />\[<br />\left( \frac{20}{x} + \frac{15}{y} \right) - \left( \frac{9}{x} + \frac{15}{y} \right) = -85 - 3<br />\]<br /><br />This simplifies to:<br /><br />\[<br />\frac{11}{x} = -88<br />\]<br /><br />Solving for \(x\):<br /><br />\[<br />x = \frac{11}{-88} = -\frac{1}{8}<br />\]<br /><br />Now, substitute \(x = -\frac{1}{8}\) back into the first original equation:<br /><br />\[<br />\frac{3}{-\frac{1}{8}} + \frac{5}{y} = 1<br />\]<br /><br />This simplifies to:<br /><br />\[<br />-24 + \frac{5}{y} = 1<br />\]<br /><br />Solving for \(\frac{5}{y}\):<br /><br />\[<br />\frac{5}{y} = 25<br />\]<br /><br />Thus,<br /><br />\[<br />y = \frac{5}{25} = \frac{1}{5}<br />\]<br /><br />So, the solution to the system of equations is:<br /><br />\[<br />x = -\frac{1}{8}, \quad y = \frac{1}{5}<br />\]