Alright, let's solve the given equation [tex]\(2x + 3y = 6\)[/tex] and express it in slope-intercept form, which is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the [tex]\(y\)[/tex]-intercept.
Step-by-Step Solution:
1. Start with the given equation:
[tex]\[
2x + 3y = 6
\][/tex]
2. Solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
First, isolate the [tex]\(3y\)[/tex] term by subtracting [tex]\(2x\)[/tex] from both sides:
[tex]\[
3y = -2x + 6
\][/tex]
Next, divide every term by 3 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \left(\frac{-2}{3}\right)x + \frac{6}{3}
\][/tex]
3. Simplify the equation:
Simplify [tex]\(\frac{6}{3}\)[/tex] to get:
[tex]\[
y = \left(\frac{-2}{3}\right)x + 2
\][/tex]
4. Identify the slope and [tex]\(y\)[/tex]-intercept:
From the equation [tex]\(y = \left(\frac{-2}{3}\right)x + 2\)[/tex], we can directly read off:
- The slope (m) is [tex]\(-\frac{2}{3}\)[/tex].
- The [tex]\(y\)[/tex]-intercept (b) is 2, which corresponds to the point [tex]\((0, 2)\)[/tex] on the coordinate plane.
Final Answers:
- The equation in slope-intercept form is [tex]\( y = \left(\frac{-2}{3}\right)x + 2 \)[/tex].
- The slope of the line is [tex]\(-\frac{2}{3}\)[/tex].
- The [tex]\(y\)[/tex]-intercept of the line is [tex]\((0, 2)\)[/tex].
