Solve Simultaneous Equations: 4x - 7/y = -3 and 5x + 2/y = 7

To solve the system of equations, we will apply the substitution method:

We start with the given equations:

$4x - \frac{7}{y} = -3$ (Equation 1)

$5x + \frac{2}{y} = 7$ (Equation 2)

Let's start by solving Equation 1 for $\frac{1}{y}$:

$4x - \frac{7}{y} = -3$

Rearrange to isolate $\frac{1}{y}$:

$-\frac{7}{y} = -3 - 4x$

Multiply through by -1 to simplify:

$\frac{7}{y} = 4x + 3$

$\frac{1}{y} = \frac{4x + 3}{7}$ (Equation 3)

Now, substitute Equation 3 into Equation 2:

$5x + \frac{2}{y} = 7$

Replace $\frac{1}{y}$ from Equation 3:

$5x + 2 \left(\frac{4x + 3}{7}\right) = 7$

Multiply both sides of the equation by 7 to eliminate the fraction:

$35x + 2(4x + 3) = 49$

Expand and simplify:

$35x + 8x + 6 = 49$

$43x + 6 = 49$

Subtract 6 from both sides:

$43x = 43$

Divide by 43:

$x = 1$

Substitute $x = 1$ back into Equation 3 to solve for $y$:

$\frac{1}{y} = \frac{4(1) + 3}{7}$

$\frac{1}{y} = \frac{4 + 3}{7}$

$\frac{1}{y} = \frac{7}{7}$

$\frac{1}{y} = 1$

Therefore, $y = 1$

The solution to the system of equations is $x = 1, y = 1$.