7. ) 2x-5y=-2 3x+4y=20

To solve the system of equations<br /><br />\[<br />\begin{cases} <br />2x - 5y = -2 \\<br />3x + 4y = 20 <br />\end{cases}<br />\]<br /><br />we can use either the substitution method or the elimination method. Here, I'll use the elimination method.<br /><br />1. **Align the equations:**<br /><br /> \[<br /> 2x - 5y = -2<br /> \]<br /> \[<br /> 3x + 4y = 20<br /> \]<br /><br />2. **Eliminate one variable:** <br /> To eliminate \(x\), we can multiply the first equation by 3 and the second equation by 2 to make the coefficients of \(x\) equal:<br /><br /> \[<br /> 3(2x - 5y) = 3(-2) \quad \Rightarrow \quad 6x - 15y = -6<br /> \]<br /> \[<br /> 2(3x + 4y) = 2(20) \quad \Rightarrow \quad 6x + 8y = 40<br /> \]<br /><br />3. **Subtract the two equations:**<br /><br /> \[<br /> (6x - 15y) - (6x + 8y) = -6 - 40<br /> \]<br /> \[<br /> 6x - 15y - 6x - 8y = -46<br /> \]<br /> \[<br /> -23y = -46<br /> \]<br /><br />4. **Solve for \(y\):**<br /><br /> \[<br /> y = \frac{-46}{-23} = 2<br /> \]<br /><br />5. **Substitute \(y = 2\) back into one of the original equations to find \(x\):** <br /> Using the first equation:<br /><br /> \[<br /> 2x - 5(2) = -2<br /> \]<br /> \[<br /> 2x - 10 = -2<br /> \]<br /> \[<br /> 2x = 8<br /> \]<br /> \[<br /> x = 4<br /> \]<br /><br />The solution to the system of equations is \(x = 4\) and \(y = 2\).