To solve the system of equations using the method of opposite coefficients (also called the elimination method), we'll eliminate one of the variables by making their coefficients opposites. Let's begin with the given system of equations:

1. $x + 3y = -3$
2. $5x - y = 17$

Method 1: Eliminate $x$

We want to eliminate $x$ by making the coefficients of $x$ in both equations opposites. We can do this by multiplying the first equation by $-5$ and leaving the second equation unchanged.

$-5(x + 3y) = -5(-3)$ $-5x - 15y = 15$

Now we have the system: $-5x - 15y = 15$ $5x - y = 17$

Next, add the two equations together to eliminate $x$:

$(-5x - 15y) + (5x - y) = 15 + 17$ $0x - 16y = 32$ $-16y = 32$

Now, solve for $y$:

$y = \frac{32}{-16} = -2$

Substitute $y = -2$ back into one of the original equations to find $x$. Using the first equation:

$x + 3(-2) = -3$ $x - 6 = -3$ $x = -3 + 6 = 3$

Thus, the solution is $x = 3$ and $y = -2$.

Method 2: Eliminate $y$

Now, let's eliminate $y$ instead. We want to make the coefficients of $y$ in both equations opposites. We can do this by multiplying the second equation by $3$ so that the coefficient of $y$ in the second equation becomes $-3$:

$x + 3y = -3$ $3(5x - y) = 3(17)$ $15x - 3y = 51$

Now we have the system: $x + 3y = -3$ $15x - 3y = 51$

Next, add the two equations together to eliminate $y$:

$(x + 3y) + (15x - 3y) = -3 + 51$ $16x = 48$

Now, solve for $x$:

$x = \frac{48}{16} = 3$

Substitute $x = 3$ back into one of the original equations to find $y$. Using the first equation:

$3 + 3y = -3$ $3y = -3 - 3$ $3y = -6$ $y = \frac{-6}{3} = -2$

Thus, the solution is $x = 3$ and $y = -2$, which matches our earlier result.

Final Answer:

The solution to the system is $x = 3$ and $y = -2$.

Would you like further details or explanations?

Here are some related questions:

1. Can you solve a system of three equations with three variables using the elimination method?
2. How does the elimination method compare to the substitution method?
3. What happens if both variables are eliminated during the elimination process?
4. Can you solve systems of nonlinear equations using the elimination method?
5. How does the determinant relate to solving systems of equations?

Tip: When choosing which variable to eliminate, it's usually easier to work with equations that have smaller or simpler coefficients!