To write the given equation [tex]\(4x + 3y = -6\)[/tex] in slope-intercept form, which is [tex]\(y = mx + b\)[/tex], follow these steps:
1. Isolate the term with [tex]\(y\)[/tex] on one side of the equation:
Start with the original equation:
[tex]\[
4x + 3y = -6
\][/tex]
Subtract [tex]\(4x\)[/tex] from both sides to isolate the term with [tex]\(y\)[/tex]:
[tex]\[
3y = -4x - 6
\][/tex]
2. Solve for [tex]\(y\)[/tex]:
Divide each term by 3 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-4x}{3} - \frac{6}{3}
\][/tex]
Simplify the fractions:
[tex]\[
y = -\frac{4}{3}x - 2
\][/tex]
Now the equation is in slope-intercept form [tex]\(y = mx + b\)[/tex], where the slope [tex]\(m\)[/tex] is [tex]\(-\frac{4}{3}\)[/tex] and the y-intercept [tex]\(b\)[/tex] is [tex]\(-2\)[/tex].
To summarize:
- Slope (m): [tex]\(-\frac{4}{3}\)[/tex] or [tex]\(-1.3333333333333333\)[/tex] (approximately)
- Y-intercept (b): [tex]\(-2\)[/tex]
### Graphing the Equation
To graph the line [tex]\(y = -\frac{4}{3}x - 2\)[/tex]:
1. Plot the y-intercept: Start by plotting the point [tex]\((0, -2)\)[/tex] on the y-axis. This is where the line intersects the y-axis.
2. Use the slope to find another point:
The slope [tex]\(-\frac{4}{3}\)[/tex] means that for every 3 units you move to the right (positive direction of x-axis), you move 4 units down (negative direction of y-axis).
- Starting from [tex]\((0, -2)\)[/tex], move 3 units to the right: [tex]\((3, -2)\)[/tex].
- From [tex]\((3, -2)\)[/tex], move 4 units down: [tex]\((3, -2 - 4) = (3, -6)\)[/tex].
Plot the point [tex]\((3, -6)\)[/tex].
3. Draw the line: Draw a straight line through the points [tex]\((0, -2)\)[/tex] and [tex]\((3, -6)\)[/tex].
Your graph should show a straight line passing through these two points with a slope of [tex]\(-\frac{4}{3}\)[/tex]. This line represents the equation [tex]\(y = -\frac{4}{3}x - 2\)[/tex].
