Let's go through each part of the system of equations given in the image and solve them.

Problem a:

We have the system of equations: $3x - 2y = 14$ $-2x + 2y = -10$

Step 1: Add the two equations to eliminate $y$: $(3x - 2y) + (-2x + 2y) = 14 + (-10)$ Simplifying: $3x - 2x = 4 \quad \Rightarrow \quad x = 4$

Step 2: Substitute $x = 4$ into the first equation: $3(4) - 2y = 14 \quad \Rightarrow \quad 12 - 2y = 14$ Solving for $y$: $-2y = 14 - 12 = 2 \quad \Rightarrow \quad y = -1$

Thus, the solution for problem (a) is $x = 4$ and $y = -1$.

Problem b:

We have the system of equations: $y = 5x + 3$ $-2x - 4y = 10$

Step 1: Substitute $y = 5x + 3$ into the second equation: $-2x - 4(5x + 3) = 10$ Simplify: $-2x - 20x - 12 = 10 \quad \Rightarrow \quad -22x - 12 = 10$ Solving for $x$: $-22x = 10 + 12 = 22 \quad \Rightarrow \quad x = -1$

Step 2: Substitute $x = -1$ into the first equation: $y = 5(-1) + 3 = -5 + 3 = -2$

Thus, the solution for problem (b) is $x = -1$ and $y = -2$.

Question c:

Which system is most efficiently solved by the Substitution Method?

System b is more efficiently solved by substitution because one of the equations ($y = 5x + 3$) is already solved for $y$. Substitution is convenient when one of the equations is in terms of a single variable.

Question d:

Which system is most efficiently solved by the Elimination Method?

System a is more efficiently solved by elimination because when the two equations are added together, the $y$ terms cancel out naturally. Elimination is useful when adding or subtracting the equations easily removes one variable, as was the case here.

Would you like further explanation on any of these steps, or should I expand on any of the methods? Here are five questions that expand on this topic:

1. What are the advantages of the substitution method for solving systems of equations?
2. How do you determine whether to use substitution or elimination for solving a system?
3. Can systems of equations have more than one solution? If so, when?
4. How can systems of equations be solved graphically, and what does the solution represent?
5. How would you modify the elimination method if the coefficients are not easily eliminated?

Tip: In elimination, multiplying one or both equations by a constant can help align the coefficients for easier elimination of a variable.