To graph the equation [tex]\(3x + 2y = 18\)[/tex], we need to find the [tex]\(x\)[/tex]-intercept, the [tex]\(y\)[/tex]-intercept, and the slope of the line. Let’s go through each step in detail.
### Finding the [tex]\(x\)[/tex]-Intercept:
The [tex]\(x\)[/tex]-intercept occurs where the graph of the equation crosses the [tex]\(x\)[/tex]-axis. At this point, [tex]\(y = 0\)[/tex]. We substitute [tex]\(y = 0\)[/tex] into the equation to find [tex]\(x\)[/tex].
[tex]\[ 3x + 2(0) = 18 \][/tex]
[tex]\[ 3x = 18 \][/tex]
[tex]\[ x = \frac{18}{3} \][/tex]
[tex]\[ x = 6 \][/tex]
So, the [tex]\(x\)[/tex]-intercept is [tex]\((6, 0)\)[/tex].
### Finding the [tex]\(y\)[/tex]-Intercept:
The [tex]\(y\)[/tex]-intercept occurs where the graph of the equation crosses the [tex]\(y\)[/tex]-axis. At this point, [tex]\(x = 0\)[/tex]. We substitute [tex]\(x = 0\)[/tex] into the equation to find [tex]\(y\)[/tex].
[tex]\[ 3(0) + 2y = 18 \][/tex]
[tex]\[ 2y = 18 \][/tex]
[tex]\[ y = \frac{18}{2} \][/tex]
[tex]\[ y = 9 \][/tex]
So, the [tex]\(y\)[/tex]-intercept is [tex]\((0, 9)\)[/tex].
### Finding the Slope:
To find the slope, we need to rewrite the equation in slope-intercept form, which is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope, and [tex]\(b\)[/tex] represents the [tex]\(y\)[/tex]-intercept.
Starting with the original equation:
[tex]\[ 3x + 2y = 18 \][/tex]
We solve for [tex]\(y\)[/tex]:
[tex]\[ 2y = -3x + 18 \][/tex]
[tex]\[ y = \frac{-3}{2}x + \frac{18}{2} \][/tex]
[tex]\[ y = -\frac{3}{2}x + 9 \][/tex]
From this, we can see that the slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex].
### Summary:
- The [tex]\(x\)[/tex]-intercept is [tex]\((6, 0)\)[/tex].
- The [tex]\(y\)[/tex]-intercept is [tex]\((0, 9)\)[/tex].
- The slope of the line is [tex]\(-\frac{3}{2}\)[/tex].
By using these points and the slope, you can graph the line representing the equation [tex]\(3x + 2y = 18\)[/tex].
