Question

Graph a line with a slope of

\[\dfrac{1}4\] that contains the point
\[(6,3)\].

Answer 1

To graph a line with a slope of \(\frac{1}{4}\) that passes through the point \((6, 3)\), we can use the point-slope form of a linear equation. The point-slope form is given by:

\[
y - y_1 = m(x - x_1)
\]

where \(m\) is the slope, and \((x_1, y_1)\) is the point the line passes through. Here, \(m = \frac{1}{4}\) and \((x_1, y_1) = (6, 3)\).

Plugging in these values, we have:

\[
y - 3 = \frac{1}{4}(x - 6)
\]

Now, let’s simplify this equation:

1. Distribute the slope on the right side:
\[
y - 3 = \frac{1}{4}x - \frac{6}{4}
\]
\[
y - 3 = \frac{1}{4}x - \frac{3}{2}
\]

2. Add 3 to both sides:
\[
y = \frac{1}{4}x - \frac{3}{2} + 3
\]
Since \(3\) can be expressed as \(\frac{6}{2}\), we can combine the constant terms:
\[
y = \frac{1}{4}x - \frac{3}{2} + \frac{6}{2}
\]
\[
y = \frac{1}{4}x + \frac{3}{2}
\]

Now we have the equation of the line in slope-intercept form:

\[
y = \frac{1}{4}x + \frac{3}{2}
\]

### Graphing the Line

1. **Plot the point** \((6, 3)\) on the graph.

2. **Find another point** using the slope. The slope \(\frac{1}{4}\) means that for every 4 units you move to the right (positive x-direction), the line moves 1 unit up (positive y-direction).

Starting from the point \((6, 3)\):
- Move 4 units to the right: \(6 + 4 = 10\)
- Move 1 unit up: \(3 + 1 = 4\)
- So, the next point is \((10, 4)\).

3. **Plot the second point** \((10, 4)\).

4. **Draw the line** through the points \((6, 3)\) and \((10, 4)\).

Now you should have a straight line passing through the point \((6, 3)\) with a slope of \(\frac{1}{4}\). The line will continue indefinitely in both directions.