To graph the equation [tex]\(4x - 3y = 12\)[/tex], follow these steps to convert it into the slope-intercept form, identify key features, and plot it accurately.
1. Start with the given equation:
[tex]\[
4x - 3y = 12
\][/tex]
2. Rearrange the equation to isolate [tex]\( y \)[/tex]:
[tex]\[
-3y = -4x + 12
\][/tex]
Dividing every term by [tex]\(-3\)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{4}{3}x - 4
\][/tex]
3. Identify the slope and the y-intercept:
The equation is now in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- The slope ([tex]\( m \)[/tex]) is [tex]\(\frac{4}{3}\)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(-4\)[/tex].
4. Plot the y-intercept:
Start by plotting the point where the line crosses the y-axis (0, -4).
5. Use the slope to determine another point:
The slope [tex]\(\frac{4}{3}\)[/tex] means that for every increase of 3 units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 4 units.
- From the y-intercept (0, -4), move right by 3 units (to [tex]\( x = 3 \)[/tex]) and up by 4 units (to [tex]\( y = 0 \)[/tex]). This gives another point (3, 0).
6. Draw the line:
Using a ruler, draw a straight line through the points (0, -4) and (3, 0).
After following these steps, the graph of the equation [tex]\(4x - 3y = 12\)[/tex] is a straight line that crosses the y-axis at [tex]\(-4\)[/tex] and has a slope of [tex]\(\frac{4}{3}\)[/tex]. This means if you start from the intercept, for every 3 units you move horizontally to the right, you move 4 units up vertically.
