Let's analyze and solve the problems from the uploaded image.

Problem Breakdown:

1.
Leon solves a system of equations and finds the solution $(-1, 3)$. Abdul solves a different system and states that their systems are equivalent.
What does this mean?
This means that Abdul’s system will also have the same solution $(-1, 3)$ because equivalent systems of equations have the same set of solutions.

2.
Why is System A equivalent to System B without solving or graphing?

System A:
$2x + 1.5y = -9$
$y = x - 6$

System B:
$3x + 0.5y = -3$
$y = x - 6$

The equations are equivalent because:

• The second equation in both systems is identical: $y = x - 6$.
• The first equation of System B can be derived by multiplying the first equation of System A by $\frac{3}{2}$.
  Verify: $\frac{3}{2} \times (2x + 1.5y = -9) \implies 3x + 0.5y = -3$ Thus, the two systems are equivalent.

3a. Solve System A using substitution.

System A: $2x + 1.5y = -9$ $y = x - 6$

Substitute $y = x - 6$ into $2x + 1.5y = -9$: $2x + 1.5(x - 6) = -9$ $2x + 1.5x - 9 = -9$ $3.5x = 0 \implies x = 0$

Substitute $x = 0$ into $y = x - 6$: $y = 0 - 6 = -6$

Solution: $(0, -6)$.

3b. Solve System B by graphing.

System B: $3x + 0.5y = -3$ $y = x - 6$

Rewriting $3x + 0.5y = -3$ in slope-intercept form ($y = mx + b$): $0.5y = -3 - 3x$ $y = -6 - 6x$

Graph the equations $y = x - 6$ and $y = -6 - 6x$ on the coordinate plane. The solution is the point where they intersect, which should match the solution $(0, -6)$.

3c. Compare the solutions.
The solutions for both systems are the same, confirming that the systems are equivalent. This validates the earlier explanation in Problem 2.

Do you want detailed graphing steps or have further questions about the solutions?

Questions to expand understanding:

1. Why do equivalent systems always share the same solutions?
2. How can you verify if two systems are equivalent using substitution or elimination?
3. Can equivalent systems have different forms of equations but the same graph?
4. How does graphing help to visually confirm a system's solution?
5. Why does scaling an equation (e.g., multiplying by a constant) not change its solution?

Tip:

When working with equivalent systems, always check if one equation is a scaled or rearranged version of another to avoid redundant solving!