The equation $3x + 3y = 3$ is a linear equation with two variables, $x$ and $y$. To simplify it, you can divide every term by 3:

$\frac{3x}{3} + \frac{3y}{3} = \frac{3}{3}$

This simplifies to:

$x + y = 1$

This is the simplified form of the original equation. Now, let's explore it:

Solutions to the Equation

The equation $x + y = 1$ also has infinitely many solutions. Here are a few examples:

1. If $x = 0$, then $y = 1$ (since $0 + y = 1$).
2. If $x = 1$, then $y = 0$ (since $1 + y = 1$).
3. If $x = -1$, then $y = 2$ (since $-1 + y = 1$).
4. If $x = 0.5$, then $y = 0.5$ (since $0.5 + y = 1$).
5. If $x = 2$, then $y = -1$ (since $2 + y = 1$).

In general, for any value of $x$, you can find $y$ by rearranging the equation to $y = 1 - x$.

Graphing the Equation

This equation represents a straight line on a graph. The slope-intercept form of the equation is:

$y = -x + 1$

• Slope (m): The slope of the line is $-1$, indicating the line is decreasing (sloping downwards) at a 45-degree angle.
• Y-intercept (b): The y-intercept is $1$, meaning the line crosses the y-axis at the point $(0, 1)$.

Would you like more details or have any questions about this?

Here are some related questions:

1. What are the x-intercept and y-intercept of the line $x + y = 1$?
2. How does changing the slope of a line affect its graph?
3. What is the relationship between two lines that have the same slope?
4. How can you determine if two lines are parallel or perpendicular?
5. What happens to the graph if the equation is modified to $2x + 2y = 4$?

Tip: Always look for ways to simplify equations by factoring out common terms. This can make solving and graphing the equation much easier.