Solving Simultaneous Equations with Fractions: Substitution Method Challenge

To solve the system of equations using the substitution method, follow these detailed steps:

We start with the given system:

$\begin{cases} \frac{-x+3y}{6}+\frac{5x-y}{3} = 12 \\ \frac{2x-4y}{10}-\frac{-2x-2y}{5} = 15 \end{cases}$

Step 1: Clear Fractions

• For the first equation, multiply through by 6 (the least common multiple of 6 and 3):
$(-x + 3y) + 2(5x - y) = 72$ $-x + 3y + 10x - 2y = 72$ $9x + y = 72 \quad \text{(Equation 1)}$
• For the second equation, multiply through by 10 (the LCM of 10 and 5):
$\frac{2x-4y}{10} \times 10 - \frac{-2x-2y}{5} \times 10 = 150$ $(2x - 4y) + (-4x - 4y) = 150$ $2x - 4y + 4x + 4y = 150$ $6x = 150$ $x = 25 \quad \text{(Equation 2)}$

Step 2: Substitute & Solve

Since we have $x = 25$ from Equation 2, substitute $x = 25$ into Equation 1:

$9(25) + y = 72$ $225 + y = 72$ $y = 72 - 225$ $y = -153$

Therefore, the solution to the system is $\boxed{x = 25, y = -153}$.