To graph the equation [tex]\( -9x + 6y = 18 \)[/tex], we first need to rearrange it into the 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's the step-by-step process:
1. Isolate [tex]\( y \)[/tex]:
[tex]\[
-9x + 6y = 18
\][/tex]
Add [tex]\( 9x \)[/tex] to both sides to get:
[tex]\[
6y = 9x + 18
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
Divide each term by 6:
[tex]\[
y = \frac{9}{6}x + \frac{18}{6}
\][/tex]
Simplify the fractions:
[tex]\[
y = 1.5x + 3
\][/tex]
Now we have the equation in the slope-intercept form:
[tex]\[
y = 1.5x + 3
\][/tex]
Here, [tex]\( m = 1.5 \)[/tex] (the slope) and [tex]\( b = 3 \)[/tex] (the y-intercept).
3. Plot the y-intercept:
Start by plotting the point where the line crosses the y-axis (i.e., where [tex]\( x = 0 \)[/tex]):
[tex]\[
\text{When } x = 0, \quad y = 3
\][/tex]
So, the point is [tex]\( (0, 3) \)[/tex].
4. Use the slope to find another point:
The slope [tex]\( m = 1.5 \)[/tex] means that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 1.5 units.
Starting from the y-intercept [tex]\( (0, 3) \)[/tex]:
[tex]\[
\text{When } x = 1, \quad y = 1.5 \cdot 1 + 3 = 4.5
\][/tex]
So, another point on the line is [tex]\( (1, 4.5) \)[/tex].
5. Draw the line:
Plot the points [tex]\( (0, 3) \)[/tex] and [tex]\( (1, 4.5) \)[/tex] on the coordinate plane. Then, draw a straight line passing through these points. Extend the line in both directions and add arrows to indicate that it continues infinitely.
6. Check with an additional point (optional but useful):
To verify accuracy, calculate another point:
[tex]\[
\text{When } x = -2, \quad y = 1.5 \cdot (-2) + 3 = -3 + 3 = 0
\][/tex]
So, [tex]\( (-2, 0) \)[/tex] is also a point on the line. Plot this point to ensure the line is correct.
By following these steps, you should get a graph of the equation [tex]\( -9x + 6y = 18 \)[/tex] that crosses the y-axis at [tex]\( (0, 3) \)[/tex] and has a slope of 1.5, meaning it rises 1.5 units for every 1 unit it runs to the right.
