Let's carefully work through question 1 step-by-step:

Problem Statement: Triangle $\triangle ABC$ is equilateral, meaning all sides are equal in length, and all angles are $60^\circ$. You are given the side lengths:

• $AB = 4x - y$
• $BC = 2x + 3y$
• $AC = 7$

Since the triangle is equilateral, we set the side lengths equal to each other:

1. Step 1: Set $AB = BC$: $4x - y = 2x + 3y$ Simplify: $4x - 2x = 3y + y$ $2x = 4y$ x = 2y \tag{1}

2. Step 2: Set $BC = AC$: $2x + 3y = 7$ Substitute $x = 2y$ from Equation (1) into this equation: $2(2y) + 3y = 7$ $4y + 3y = 7$ $7y = 7$ y = 1 \tag{2}

3. Step 3: Solve for $x$: Substitute $y = 1$ into $x = 2y$: $x = 2(1)$ x = 2 \tag{3}

Solution:

• $x = 2$
• $y = 1$

Verification:

• Check $AB = BC = AC$:
  • $AB = 4x - y = 4(2) - 1 = 8 - 1 = 7$
  • $BC = 2x + 3y = 2(2) + 3(1) = 4 + 3 = 7$
  • $AC = 7$
  • Since $AB = BC = AC = 7$, the solution is correct.

Let me know if you have any questions or would like more details. Here are five follow-up questions to expand on this:

1. How do we know that equilateral triangles have equal side lengths?
2. What is the significance of substituting one variable in terms of the other?
3. Can the method used here be generalized for all equilateral triangles?
4. What happens if the triangle is isosceles instead of equilateral?
5. How would the solution change if one side length had a different equation?

Tip: Always verify your solution by substituting the values back into the original equations!