The given equation $2x + 3y = 6$ is a linear equation in two variables ($x$ and $y$). Let’s analyze and solve for different scenarios:

1. Solving for $y$ in terms of $x$:

$2x + 3y = 6$ Subtract $2x$ from both sides: $3y = 6 - 2x$ Divide through by 3: $y = \frac{6 - 2x}{3} \quad \text{or} \quad y = 2 - \frac{2x}{3}$

2. Solving for $x$ in terms of $y$:

$2x + 3y = 6$ Subtract $3y$ from both sides: $2x = 6 - 3y$ Divide through by 2: $x = \frac{6 - 3y}{2} \quad \text{or} \quad x = 3 - \frac{3y}{2}$

3. Finding intercepts:

• $x$-intercept: Set $y = 0$: $2x + 3(0) = 6 \implies x = 3$ So, the $x$-intercept is $(3, 0)$.

• $y$-intercept: Set $x = 0$: $2(0) + 3y = 6 \implies y = 2$ So, the $y$-intercept is $(0, 2)$.

4. Graph of the Equation:

The equation represents a straight line. Using the intercepts $(3, 0)$ and $(0, 2)$, we can draw the line on a Cartesian plane.

5. General Observations:

This equation represents a family of solutions that can be written as $(x, y)$ pairs satisfying the equation. For example:

• If $x = 0$, $y = 2$.
• If $x = 3$, $y = 0$.
• If $x = 1$, substitute into $y = 2 - \frac{2x}{3}$: $y = 2 - \frac{2(1)}{3} = 2 - \frac{2}{3} = \frac{4}{3}$ So, $(1, \frac{4}{3})$ is also a solution.

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Do you want further details, such as graphing this equation or exploring specific solutions?

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Expanding Your Understanding:

1. What are the solutions if this equation is paired with $x - y = 1$?
2. How does this equation change if transformed into slope-intercept form ($y = mx + b$)?
3. What is the slope of this line, and what does it represent geometrically?
4. How do you test whether a point (e.g., $(1, 1)$) satisfies this equation?
5. How would this equation change if it were written in standard form $Ax + By = C$ with $A > 0$?

Tip: Always check intercepts first when graphing a linear equation—they provide a quick way to draw the line!