To graph the equation [tex]\( -3x + 4y = -12 \)[/tex], it is often helpful to rewrite 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 y-intercept.
Here are the steps to convert the given equation to slope-intercept form:
1. Start with the given equation:
[tex]\[
-3x + 4y = -12
\][/tex]
2. Isolate the term involving [tex]\( y \)[/tex] on one side:
[tex]\[
4y = 3x - 12
\][/tex]
3. Solve for [tex]\( y \)[/tex] by dividing every term by 4:
[tex]\[
y = \frac{3x}{4} - 3
\][/tex]
So, the equation in slope-intercept form is:
[tex]\[
y = \frac{3x}{4} - 3
\][/tex]
Now, let's interpret this equation:
- The slope ([tex]\( m \)[/tex]) of the line is [tex]\( \frac{3}{4} \)[/tex]. This means that for every 4 units you move horizontally to the right, you move 3 units up vertically.
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(-3\)[/tex]. This is where the line crosses the y-axis.
### To graph the equation:
1. Plot the y-intercept:
- Start by plotting the point [tex]\((0, -3)\)[/tex] on the y-axis.
2. Use the slope to find another point:
- From the y-intercept [tex]\((0, -3)\)[/tex], use the slope [tex]\(\frac{3}{4}\)[/tex].
- This means from [tex]\((0, -3)\)[/tex], move 4 units to the right (positive direction on the x-axis) and 3 units up (positive direction on the y-axis).
- This will get you to the point [tex]\((4, 0)\)[/tex].
3. Draw the line:
- Once you have these two points, [tex]\((0, -3)\)[/tex] and [tex]\((4, 0)\)[/tex], draw a straight line through these points. This line represents the graph of the equation [tex]\( -3x + 4y = -12 \)[/tex].
Thus, the graph of the equation [tex]\( -3x + 4y = -12 \)[/tex] is a straight line passing through the points [tex]\((0, -3)\)[/tex] and [tex]\((4, 0)\)[/tex] with a slope of [tex]\( \frac{3}{4} \)[/tex].
