Solve the Fractional System: 1/3x - 4y = 5 and x + 6y = 9

To solve this system of equations, we are going to use the substitution method:

Given the equations:

$\begin{cases} \frac{1}{3}x - 4y = 5 \quad \text{(Equation 1)} \\ x + 6y = 9 \quad \text{(Equation 2)} \end{cases}$

• First, we solve Equation 2 for $x$:

$x = 9 - 6y$

• Substitute this expression for $x$ into Equation 1:

$\frac{1}{3}(9 - 6y) - 4y = 5$

Multiply through by 3 to eliminate fractions:

$9 - 6y - 12y = 15$

Combine like terms:

$9 - 18y = 15$

Subtract 9 from both sides:

$-18y = 6$

Divide both sides by -18:

$y = -\frac{1}{3}$

• Substitute $y = -\frac{1}{3}$ back into the expression for $x$ from Equation 2:

$x = 9 - 6(-\frac{1}{3})$

$x = 9 + 2$

$x = 11$

Thus, the solution to the system of equations is:

$x = 11, y = -\frac{1}{3}$.