Sure! Let's use the slope-intercept form to graph the given equation [tex]\(3x + 2y = 6\)[/tex].
### Step-by-Step Solution:
1. Understand the Given Equation:
The equation provided is in standard form: [tex]\(3x + 2y = 6\)[/tex].
2. Convert to the Slope-Intercept Form:
To convert the equation to the slope-intercept form, which is [tex]\(y = mx + b\)[/tex] where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept, we need to solve for [tex]\(y\)[/tex]:
[tex]\[
3x + 2y = 6
\][/tex]
First, isolate [tex]\(2y\)[/tex] on one side of the equation:
[tex]\[
2y = -3x + 6
\][/tex]
Next, divide every term by 2:
[tex]\[
y = \frac{-3}{2}x + 3
\][/tex]
3. Identify the Slope and Y-Intercept:
From the slope-intercept form [tex]\(y = \frac{-3}{2}x + 3\)[/tex], we identify:
- The slope ([tex]\(m\)[/tex]) is [tex]\(-\frac{3}{2}\)[/tex].
- The y-intercept ([tex]\(b\)[/tex]) is 3.
4. Graphing the Equation:
- Plot the Y-Intercept: Start by plotting the y-intercept on the graph. Since [tex]\(b = 3\)[/tex], place a point at [tex]\((0, 3)\)[/tex] on the y-axis.
- Use the Slope to Find Another Point: The slope is [tex]\(-\frac{3}{2}\)[/tex], which means for every 2 units you move to the right (positive direction of the x-axis), you move 3 units down (negative direction of the y-axis).
From the y-intercept point [tex]\((0, 3)\)[/tex]:
- Move 2 units to the right to [tex]\((2, 3)\)[/tex].
- Then move 3 units down to [tex]\((2, 0)\)[/tex].
- Draw the Line: With the points [tex]\((0, 3)\)[/tex] and [tex]\((2, 0)\)[/tex] plotted, draw a straight line through these points to represent the equation [tex]\(y = \frac{-3}{2}x + 3\)[/tex].
### Conclusion
To sum up, the graph of the equation [tex]\(3x + 2y = 6\)[/tex] is a straight line that crosses the y-axis at 3 (the y-intercept) and has a slope of [tex]\(-\frac{3}{2}\)[/tex].
