To write the line in slope-intercept form ([tex]\(y = mx + b\)[/tex]) and graph it on the coordinate plane, we will follow these steps:
1. Rewrite the equation in slope-intercept form:
The given equation is [tex]\( 4x + 3y = -6 \)[/tex].
To convert this into slope-intercept form, [tex]\( y = mx + b \)[/tex], we need to solve for [tex]\( y \)[/tex].
- Start by isolating the [tex]\( y \)[/tex]-term on one side of the equation:
[tex]\[
3y = -4x - 6
\][/tex]
- Now, divide every term by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-4}{3}x - 2
\][/tex]
The equation in slope-intercept form is:
[tex]\[
y = -\frac{4}{3}x - 2
\][/tex]
Here, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{4}{3} \)[/tex] and the y-intercept [tex]\( b \)[/tex] is [tex]\( -2 \)[/tex].
2. Identify the slope and y-intercept:
- The slope [tex]\( m \)[/tex] is [tex]\( -1.3333333333333333 \)[/tex] (equivalent to [tex]\( -\frac{4}{3} \)[/tex]).
- The y-intercept [tex]\( b \)[/tex] is [tex]\( -2 \)[/tex].
3. Graph the line on the coordinate plane:
To graph the equation [tex]\( y = -\frac{4}{3}x - 2 \)[/tex]:
- Plot the y-intercept: The y-intercept is the point where the line crosses the y-axis. For [tex]\( b = -2 \)[/tex], plot the point [tex]\((0, -2)\)[/tex] on the coordinate plane.
- Use the slope to find another point:
- The slope [tex]\( m = -\frac{4}{3} \)[/tex] tells us that for every 3 units you move to the right (positive x direction), you move 4 units down (negative y direction).
- Starting from the y-intercept [tex]\((0, -2)\)[/tex], move 3 units to the right to [tex]\((3, -2)\)[/tex], and then move 4 units down to [tex]\((3, -6)\)[/tex]. Plot the point [tex]\((3, -6)\)[/tex].
- Draw the line: Connect the points [tex]\((0, -2)\)[/tex] and [tex]\((3, -6)\)[/tex] with a straight line. This line represents the equation [tex]\( y = -\frac{4}{3}x - 2 \)[/tex].
By following these steps, you have graphed the line [tex]\( 4x + 3y = -6 \)[/tex] in slope-intercept form on the coordinate plane.
