Let's solve the given system of equations step by step:

The system of equations provided is:

$-\frac{3}{5}x + 5y = -\frac{1}{5}$ $\frac{1}{3}x - \frac{1}{2}y = -9$

Step 1: Eliminate fractions in the equations

To simplify calculations, eliminate fractions by multiplying through by the least common denominator (LCD) in each equation.

1. First equation: Multiply through by 5 (the LCD of the terms): $5 \times \left(-\frac{3}{5}x + 5y\right) = 5 \times -\frac{1}{5}$ This simplifies to: $-3x + 25y = -1$

2. Second equation: Multiply through by 6 (the LCD of $\frac{1}{3}$ and $\frac{1}{2}$): $6 \times \left(\frac{1}{3}x - \frac{1}{2}y\right) = 6 \times -9$ This simplifies to: $2x - 3y = -54$

Now the system becomes: $-3x + 25y = -1 \quad \text{(1)}$ $2x - 3y = -54 \quad \text{(2)}$

Step 2: Solve the system of equations

We will use the elimination or substitution method to solve.

Elimination method:

Multiply both equations to make the coefficients of $x$ equal:

• Multiply equation (1) by 2: $-6x + 50y = -2 \quad \text{(3)}$
• Multiply equation (2) by 3: $6x - 9y = -162 \quad \text{(4)}$

Add equations (3) and (4) together: $(-6x + 6x) + (50y - 9y) = -2 - 162$ $41y = -164$ $y = -4$

Substitute $y = -4$ into one of the original equations:

Using equation (2): $2x - 3y = -54$ $2x - 3(-4) = -54$ $2x + 12 = -54$ $2x = -66$ $x = -33$

Final Answer:

$x = -33, \quad y = -4$

Would you like a detailed verification or further breakdown? Let me know!

Five Related Questions:

1. How do we verify solutions to a system of linear equations?
2. Can we solve this problem using matrices and determinants?
3. What is the graphical representation of this system?
4. How do we simplify calculations with fractions in equations?
5. Why might substitution or elimination work better in certain cases?

Tip:

When working with fractions, clearing the denominators at the start simplifies calculations and minimizes errors.