Let's find the equation of the line given the information provided:
1. Identify the y-intercept:
- The line intercepts the [tex]\( y \)[/tex]-axis at [tex]\(-3\)[/tex]. This means that the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is [tex]\(-3\)[/tex].
2. Determine the slope:
- We know that for every 2 units increase in the [tex]\( y \)[/tex]-coordinate, the [tex]\( x \)[/tex]-coordinate decreases by 7 units.
- The slope ([tex]\( m \)[/tex]) of the line is the change in [tex]\( y \)[/tex] divided by the change in [tex]\( x \)[/tex]. Here, the change in [tex]\( y \)[/tex] ([tex]\( \Delta y \)[/tex]) is 2, and the change in [tex]\( x \)[/tex] ([tex]\( \Delta x \)[/tex]) is -7. Thus, the slope [tex]\( m \)[/tex] is calculated as follows:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{2}{-7} = -\frac{2}{7}
\][/tex]
3. Write the equation of the line:
- The general form of the equation of a line is:
[tex]\[
y = mx + b
\][/tex]
- Substituting the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] into the equation:
[tex]\[
y = -\frac{2}{7}x - 3
\][/tex]
Therefore, the correct equation for this line is:
[tex]\[
\boxed{y = -\frac{2}{7} x - 3}
\][/tex]
So, the correct answer choice is:
[tex]\[
\text{(C) } y = -\frac{2}{7} x - 3
\][/tex]
