Solve the system of equations. 2x-3y=1 4x-5y=5 x= square y= square

We are tasked with solving the system of linear equations:<br /><br />\[<br />\begin{aligned}<br />1) & \quad 2x - 3y = 1, \\<br />2) & \quad 4x - 5y = 5.<br />\end{aligned}<br />\]<br /><br />### Step 1: Solve one equation for one variable<br />From equation (1), solve for \(x\) in terms of \(y\):<br />\[<br />2x - 3y = 1 \implies 2x = 3y + 1 \implies x = \frac{3y + 1}{2}.<br />\]<br /><br />### Step 2: Substitute into the second equation<br />Substitute \(x = \frac{3y + 1}{2}\) into equation (2):<br />\[<br />4x - 5y = 5.<br />\]<br />Replace \(x\) with \(\frac{3y + 1}{2}\):<br />\[<br />4\left(\frac{3y + 1}{2}\right) - 5y = 5.<br />\]<br />Simplify:<br />\[<br />2(3y + 1) - 5y = 5 \implies 6y + 2 - 5y = 5.<br />\]<br />Combine like terms:<br />\[<br />y + 2 = 5 \implies y = 3.<br />\]<br /><br />### Step 3: Solve for \(x\)<br />Substitute \(y = 3\) into the expression for \(x\):<br />\[<br />x = \frac{3y + 1}{2} \implies x = \frac{3(3) + 1}{2} \implies x = \frac{9 + 1}{2} \implies x = \frac{10}{2} \implies x = 5.<br />\]<br /><br />### Final Answer:<br />\[<br />x = 5, \quad y = 3.<br />\]