6. Penulutis cincremy ypannemiti ) 2x-3y=3 x+5y=7

To solve the system of equations:<br /><br />\[<br />\begin{cases}<br />2x - 3y = 3 \\<br />x + 5y = 7<br />\end{cases}<br />\]<br /><br />we can use the substitution or elimination method. Here, we'll use the elimination method.<br /><br />First, let's multiply the second equation by 2 to align the coefficients of \(x\):<br /><br />\[<br />2(x + 5y) = 2 \cdot 7 \implies 2x + 10y = 14<br />\]<br /><br />Now we have the system:<br /><br />\[<br />\begin{cases}<br />2x - 3y = 3 \\<br />2x + 10y = 14<br />\end{cases}<br />\]<br /><br />Next, we subtract the first equation from the second to eliminate \(x\):<br /><br />\[<br />(2x + 10y) - (2x - 3y) = 14 - 3<br />\]<br /><br />Simplify the left side:<br /><br />\[<br />2x + 10y - 2x + 3y = 11 \implies 13y = 11<br />\]<br /><br />Solve for \(y\):<br /><br />\[<br />y = \frac{11}{13}<br />\]<br /><br />Now, substitute \(y = \frac{11}{13}\) back into the second original equation \(x + 5y = 7\):<br /><br />\[<br />x + 5 \left(\frac{11}{13}\right) = 7<br />\]<br /><br />Simplify:<br /><br />\[<br />x + \frac{55}{13} = 7<br />\]<br /><br />Convert 7 to a fraction with a denominator of 13:<br /><br />\[<br />x + \frac{55}{13} = \frac{91}{13}<br />\]<br /><br />Subtract \(\frac{55}{13}\) from both sides:<br /><br />\[<br />x = \frac{91}{13} - \frac{55}{13} = \frac{36}{13}<br />\]<br /><br />So, the solution to the system of equations is:<br /><br />\[<br />x = \frac{36}{13}, \quad y = \frac{11}{13}<br />\]