The equation $x + 3y = -9$ is a linear equation in two variables, $x$ and $y$. To analyze it, let’s explore a few possible steps:

1. Rewriting the Equation

You can rewrite it in terms of one variable. For instance:

• Solve for $x$ in terms of $y$:
  $x = -9 - 3y$

• Solve for $y$ in terms of $x$:
  $y = \frac{-9 - x}{3}$

2. Finding Intercepts

• x-intercept (set $y = 0$):
  $x + 3(0) = -9 \implies x = -9$

• y-intercept (set $x = 0$):
  $0 + 3y = -9 \implies y = -3$

Thus, the intercepts are $(-9, 0)$ and $(0, -3)$.

3. Graphing the Line

The equation represents a straight line. To graph:

• Plot the intercepts $(-9, 0)$ and $(0, -3)$.
• Draw a straight line through these points.

4. General Form and Slope-Intercept Form

• General form: $x + 3y = -9$
• Slope-intercept form: Rewrite the equation as $y = mx + b$: $y = -\frac{1}{3}x - 3$ Here, the slope ($m$) is $-\frac{1}{3}$, and the y-intercept is $-3$.

Would you like me to solve for specific values, graph it, or analyze further?

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Related Questions:

1. How can we find additional points that satisfy $x + 3y = -9$?
2. What is the slope and direction of the line $x + 3y = -9$?
3. How would the graph change if the equation was $x + 3y = c$, where $c$ is any constant?
4. How can you verify if a point lies on the line $x + 3y = -9$?
5. Can we use this equation to form a system of equations with another line? What are the possible outcomes?

Tip:

For equations in the form $ax + by = c$, rewriting them into slope-intercept form $y = mx + b$ makes it easier to understand their slope and graph them.