To solve the system of equations using the elimination method:

We are given:

1. $3x + y = 8$
2. $5x + y = 10$

Step 1: Eliminate $y$

To eliminate $y$, subtract the first equation from the second equation.

$(5x + y) - (3x + y) = 10 - 8$

Simplify: $(5x - 3x) + (y - y) = 2$ $2x = 2$

Step 2: Solve for $x$

Divide both sides of the equation by 2: $x = 1$

Step 3: Substitute $x = 1$ into one of the original equations

Let's use the first equation: $3x + y = 8$ Substitute $x = 1$: $3(1) + y = 8$ $3 + y = 8$ $y = 8 - 3$ $y = 5$

Final Answer:

The solution to the system is: $x = 1, \quad y = 5$

Would you like further details or explanations?

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Here are 5 related questions to explore:

1. How can substitution be used to solve the same system of equations?
2. What happens if we multiply the equations by different factors before eliminating?
3. Can the graph of these two equations help verify the solution?
4. What is the significance of a system having one solution like this?
5. How do we know if a system of equations has no solution or infinite solutions?

Tip: Always check your solution by plugging the values back into the original equations!