Let's solve the given problem step by step and show that both methods result in the same solution.

The system of equations is:

1. $x + 3y = -5$
2. $9x + 3y = 3$

Tyler's First Step: Isolate $x$ in the first equation.

Starting with $x + 3y = -5$: $x = -5 - 3y$

Substitute $x = -5 - 3y$ into the second equation $9x + 3y = 3$: $9(-5 - 3y) + 3y = 3$

Simplify: $-45 - 27y + 3y = 3$ $-45 - 24y = 3$ $-24y = 48 \quad \text{(add 45 to both sides)}$ $y = -2$

Now substitute $y = -2$ back into $x = -5 - 3y$: $x = -5 - 3(-2)$ $x = -5 + 6$ $x = 1$

Thus, the solution is: $(x, y) = (1, -2)$

Han's First Step: Isolate $3y$ in the first equation.

Starting with $x + 3y = -5$: $3y = -5 - x$

Substitute $3y = -5 - x$ into the second equation $9x + 3y = 3$: $9x + (-5 - x) = 3$

Simplify: $9x - 5 - x = 3$ $8x - 5 = 3$ $8x = 8 \quad \text{(add 5 to both sides)}$ $x = 1$

Now substitute $x = 1$ back into $3y = -5 - x$: $3y = -5 - 1$ $3y = -6$ $y = -2$

Thus, the solution is: $(x, y) = (1, -2)$

Conclusion:

Both Tyler's and Han's approaches lead to the same solution, $(x, y) = (1, -2)$.

Would you like a breakdown of any specific step? Let me know!

Related Questions:

1. Can you explain the substitution method in general?
2. How does isolating $x$ versus isolating $3y$ affect the solution process?
3. Can this system also be solved using the elimination method?
4. How can graphing the equations confirm the solution?
5. What are some common mistakes to avoid when using substitution?

Tip:

Always double-check your substitution by plugging the solution back into both original equations to ensure it satisfies both!