Solve the System: Tackling -y + 2/5x = 13 and 1/2y + 2x = 10

Solve the above set of equations and choose the correct answer.

$\begin{cases} -y+\frac{2}{5}x=13 \\ \frac{1}{2}y+2x=10 \end{cases}$

To solve the given system of equations, we follow these steps:

Given equations:

• Equation 1: $-y + \frac{2}{5}x = 13$
• Equation 2: $\frac{1}{2}y + 2x = 10$

Step 1: Clear fractions in Equation 1 by multiplying through by 5:

$-5y + 2x = 65$ ...(Equation 3)

Step 2: Clear fractions in Equation 2 by multiplying through by 2:

$y + 4x = 20$ ...(Equation 4)

Step 3: Align the coefficients of $y$ for elimination. Use Equation 3 and Equation 4, where coefficients of $y$ can be easily handled.

Using Equations 3 and 4:

$-5y + 2x = 65$

$y + 4x = 20$

Step 4: Let's multiply Equation 4 by 5 to align coefficients of $y$:

$5y + 20x = 100$

Step 5: Add the resulting Equation 4 to Equation 3:

$-5y + 2x + 5y + 20x = 65 + 100$

$22x = 165$

Step 6: Solve for $x$:

$x = \frac{165}{22} = 7.5$

Step 7: Substitute $x = 7.5$ back into Equation 4 to solve for $y$:

$y + 4(7.5) = 20$

$y + 30 = 20$

$y = 20 - 30 = -10$

Therefore, the solution is $x = 7.5 \text{ and } y = -10$.

The correct choice from the answer options is:

$x=7.5,y=-10$