https://file.aiquickdraw.com/mathbot/file/1728257065550o6by153q.jpg

The given function is:

\[
q(x) = \frac{5x - (4 + 3x)}{6}
\]

We need to simplify this expression first to find its slope-intercept form.

### Step 1: Simplifying the function
\[
q(x) = \frac{5x - (4 + 3x)}{6}
\]
Distribute the negative sign in the numerator:
\[
q(x) = \frac{5x - 4 - 3x}{6}
\]
Now, combine like terms:
\[
q(x) = \frac{(5x - 3x) - 4}{6} = \frac{2x - 4}{6}
\]
Simplify the expression:
\[
q(x) = \frac{2x}{6} - \frac{4}{6} = \frac{x}{3} - \frac{2}{3}
\]

### Step 2: Identifying the slope and y-intercept
Now, the equation is in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the simplified equation:
\[
q(x) = \frac{1}{3}x - \frac{2}{3}
\]

- Slope (\( m \)) = \( \frac{1}{3} \)
- Y-intercept = \( \left( 0, -\frac{2}{3} \right) \)

### Step 3: Finding two points to graph the line
We already know one point: the y-intercept \( (0, -\frac{2}{3}) \).

To find another point, let's substitute a value of \( x \) to find \( y \).

For \( x = 3 \):
\[
q(3) = \frac{1}{3}(3) - \frac{2}{3} = 1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}
\]
So, another point is \( (3, \frac{1}{3}) \).

### Final Answer
Two points on the line are:
1. \( (0, -\frac{2}{3}) \)
2. \( (3, \frac{1}{3}) \)

Do you want more details or have any further questions?

Here are some related questions for further exploration:
1. How can you graph a line given the slope and one point?
2. What is the general form of a linear equation and how can you convert between forms?
3. How do changes in the slope affect the steepness of the line?
4. Can you explain how to find the x-intercept from the equation \( q(x) = \frac{x}{3} - \frac{2}{3} \)?
5. How do you find the equation of a line given two points?

**Tip:** When graphing a line, it's useful to find at least two points, such as the intercepts, to ensure accuracy.

Consider the function q(x) = (5x - (4 + 3x)) / 6. Find two points on the line to graph the function.

Learn how to simplify and graph the linear function q(x) = (5x - (4 + 3x)) / 6. Discover the slope, y-intercept, and find two points on the line for accurate graphing. Suitable for students in Grades 9-10.

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