The problem in the image asks to solve the system of linear equations using the "elimination method." The system of equations provided is:

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

Step-by-step solution using the elimination method:

1. Align the equations:

We have: $3x - y = 5 \quad \text{(Equation 1)}$ $x + 3y = 5 \quad \text{(Equation 2)}$

2. Eliminate one variable:

To eliminate $y$, we'll multiply both equations to make the coefficients of $y$ equal (with opposite signs). Let's multiply Equation 2 by 3:

$3(x + 3y) = 3 \times 5$ This gives: $3x + 9y = 15 \quad \text{(Equation 3)}$

Now subtract Equation 1 from Equation 3 to eliminate $x$: $(3x + 9y) - (3x - y) = 15 - 5$ This simplifies to: $10y = 10$ $y = 1$

3. Substitute $y = 1$ back into one of the original equations:

Substitute $y = 1$ into Equation 1: $3x - 1 = 5$ $3x = 6$ $x = 2$

Final Answer:

The solution to the system is: $x = 2, \quad y = 1$

Would you like more details or have any further questions?

Here are five related questions for practice:

1. Can you solve the system of equations using the substitution method?
2. What happens if both equations have the same slope?
3. How would you solve the system if the equations were $2x - y = 7$ and $4x + 2y = 8$?
4. How can we check the solution to a system of equations graphically?
5. What is the general form of a system of linear equations in two variables?

Tip: In the elimination method, always try to make the coefficients of one variable the same (or opposites) before adding or subtracting the equations.