In this image, we are given four balance models (A, B, C, D), each representing a one-step equation. Based on the key provided, the symbols in the models translate to variables and constants as follows:

• Green bar: $x$ (positive variable, +1).
• Yellow square: $1$ (positive constant, +1).
• Red square: $-1$ (negative constant, -1).

Let's solve each model by translating the balance diagram into an equation and solving for $x$:

A

• Left side: 2 green bars $(2x)$ and 6 yellow squares $(+6)$.
• Right side: 3 yellow squares $(+3)$.

Equation: $2x + 6 = 3$

Solution: To solve for $x$: $2x = 3 - 6$ $2x = -3$ $x = \frac{-3}{2}$

B

• Left side: 3 green bars $(3x)$.
• Right side: 9 yellow squares $(+9)$.

Equation: $3x = 9$

Solution: $x = \frac{9}{3} = 3$

C

• Left side: 3 green bars $(3x)$ and 4 red squares $(-4)$.
• Right side: 1 yellow square $(+1)$.

Equation: $3x - 4 = 1$

Solution: $3x = 1 + 4$ $3x = 5$ $x = \frac{5}{3}$

D

• Left side: 2 green bars $(2x)$.
• Right side: 2 yellow squares $(+2)$.

Equation: $2x = 2$

Solution: $x = \frac{2}{2} = 1$

Would you like more details on any of these steps or explanations? Here are some follow-up questions that can deepen your understanding:

1. How would the solution change if the yellow squares were negative in any of the models?
2. How can you check if your solution for $x$ is correct in each model?
3. What if the red squares represented $-2$ instead of $-1$, how would that affect the equations?
4. Can you create similar equations using different numbers of squares and bars?
5. How do one-step equations differ from two-step equations, and what strategies apply to each?

Tip: To solve equations with variables on both sides, always try to isolate the variable on one side of the equation by using inverse operations.