Question

Select all statements that are true about the graph of 2x+3y=12

Answer 1

To analyze the graph of the equation \(2x + 3y = 12\), we can derive some important characteristics:

1. **Form of the Equation**: The equation \(2x + 3y = 12\) is in standard form, which can be rewritten in slope-intercept form (y = mx + b) for easier analysis.

Rearranging the equation:
\[
3y = -2x + 12
\]
\[
y = -\frac{2}{3}x + 4
\]

Here, the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(b\) is \(4\).

2. **Y-Intercept**: The y-intercept of the graph is the point where it crosses the y-axis. From the slope-intercept form, the y-intercept is the point \((0, 4)\).

3. **X-Intercept**: To find the x-intercept, set \(y = 0\):
\[
2x + 3(0) = 12 \implies 2x = 12 \implies x = 6
\]
The x-intercept is the point \((6, 0)\).

4. **Slope**: The slope of the line is \(-\frac{2}{3}\), which indicates that the line descends (falls) as it moves from left to right.

5. **Linear Relationship**: The equation represents a line in the Cartesian plane, reflecting a linear relationship between \(x\) and \(y\).

6. **Graph Appearance**: The line will extend infinitely in both directions, passing through the points \((0, 4)\) and \((6, 0)\).

Based on this analysis, here are possible true statements about the graph of \(2x + 3y = 12\):

- The line has a negative slope.
- The y-intercept is at \( (0, 4) \).
- The x-intercept is at \( (6, 0) \).
- This is a linear equation, and its graph is a straight line.
- The line decreases as it moves from left to right.

If you have specific statements to evaluate, please provide those for a more tailored assessment!