To solve the given system of equations using the elimination method, let's look at the equations:

1. $7x - 5y = 15$
2. $-x + 2y = -15$

Step 1: Align the Equations

The equations are already aligned as follows: $7x - 5y = 15$ $-x + 2y = -15$

Step 2: Eliminate One Variable

To eliminate $x$, we can multiply the second equation by $7$ to match the coefficient of $x$ in the first equation.

So, multiply the second equation by $7$: $-7x + 14y = -105$

Now we have: $7x - 5y = 15$ $-7x + 14y = -105$

Step 3: Add the Equations

Adding these two equations will eliminate $x$: $(7x - 5y) + (-7x + 14y) = 15 - 105$ $0x + 9y = -90$ $9y = -90$

Step 4: Solve for $y$

Divide both sides by $9$: $y = -10$

Step 5: Substitute $y$ Back to Find $x$

Now that we know $y = -10$, substitute it into one of the original equations. Let's use the second equation: $-x + 2(-10) = -15$ $-x - 20 = -15$ $-x = 5$ $x = -5$

Solution

The solution to the system is: $(x, y) = (-5, -10)$

Would you like further details on any steps or have additional questions?

Here are five related questions to further explore this concept:

1. How does the elimination method compare to the substitution method in solving systems of equations?
2. What would happen if the coefficients of $x$ or $y$ did not align as easily?
3. Can systems of equations sometimes have no solution or infinitely many solutions?
4. How can we determine which variable is easier to eliminate when solving a system?
5. What are some real-life applications where solving systems of equations is necessary?

Tip: When using elimination, always check if the coefficients can be adjusted easily by multiplication to simplify the process of adding or subtracting the equations.