We are given the system of equations:

$\text{1) } 4x - 5y = 2$ $\text{2) } 3x - y = 8$

To determine an equivalent system, we can use the elimination or substitution method. Here, we will solve it using the elimination method by eliminating one variable.

Step 1: Align both equations

$\text{1) } 4x - 5y = 2$ $\text{2) } 3x - y = 8$

Step 2: Eliminate one variable (let’s eliminate $y$)

We can multiply the second equation by 5 to align the coefficients of $y$:

$5(3x - y = 8) \Rightarrow 15x - 5y = 40$

Now the system becomes: $\text{1) } 4x - 5y = 2$ $\text{2) } 15x - 5y = 40$

Step 3: Subtract the first equation from the second

$(15x - 5y) - (4x - 5y) = 40 - 2$ $(15x - 4x) = 38$ $11x = 38$

Solve for $x$: $x = \frac{38}{11}$

Step 4: Substitute $x = \frac{38}{11}$ into one of the original equations

We will substitute it into the second equation $3x - y = 8$:

$3 \left(\frac{38}{11}\right) - y = 8$ $\frac{114}{11} - y = 8$

Convert 8 to have a denominator of 11: $\frac{114}{11} - y = \frac{88}{11}$

Now subtract $\frac{88}{11}$ from both sides: $\frac{114}{11} - \frac{88}{11} = y$ $y = \frac{26}{11}$

Final Answer:

The solution to the system is $x = \frac{38}{11}$ and $y = \frac{26}{11}$.

This means the equivalent system represents the same solution as the original.

Would you like further details, or have any other questions?

Here are 5 related questions to explore further:

1. How does the substitution method work for solving systems of equations?
2. What is the graphical interpretation of the solution to this system?
3. How can we verify that the solution is correct?
4. Can this system be solved using matrix methods like Gaussian elimination?
5. What is the determinant of the coefficient matrix for this system?

Tip: When solving a system of linear equations, elimination helps when the coefficients of one variable can be easily aligned.