Sure! Let's derive the equation of the line given the slope and the y-intercept step-by-step.
1. Identify the components of the slope-intercept form:
The general form of the equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept (the point where the line crosses the y-axis).
2. Substitute the given values:
- We are given the slope [tex]\( \frac{2}{5} \)[/tex].
- We are given the y-intercept [tex]\( 5 \)[/tex].
3. Write the equation:
Substitute [tex]\( m = \frac{2}{5} \)[/tex] and [tex]\( b = 5 \)[/tex] into the slope-intercept form:
[tex]\[
y = \frac{2}{5}x + 5
\][/tex]
So, the equation of the line is:
[tex]\[
y = 0.4x + 5
\][/tex]
Next, let's talk about how to graph this line:
- Step 1: Plot the y-intercept [tex]\( (0, 5) \)[/tex]. This is the point where the line crosses the y-axis.
- Step 2: Use the slope to determine another point on the line. The slope [tex]\( \frac{2}{5} \)[/tex] means that for every 5 units you move to the right along the x-axis, you move 2 units up along the y-axis.
Starting from the y-intercept (0, 5):
- Move 5 units to the right: [tex]\( (0 + 5, 5) = (5, 5) \)[/tex]
- Move 2 units up: [tex]\( (5, 5 + 2) = (5, 7) \)[/tex]
So, another point on the line is [tex]\( (5, 7) \)[/tex].
- Step 3: Draw the line through the points [tex]\( (0, 5) \)[/tex] and [tex]\( (5, 7) \)[/tex].
This will give you the graph of the line corresponding to the equation [tex]\( y = 0.4x + 5 \)[/tex].
