Sure, let's discuss the equation of the line given as \( y = -2x - 2 \).
This equation is already in the slope-intercept form, which is \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept. Let's break it down step-by-step:
1. Identify the slope (m):
The slope (m) of the line is the coefficient of \( x \) in the equation. In the given equation:
[tex]\[
y = -2x - 2
\][/tex]
The coefficient of \( x \) is \( -2 \). Hence, the slope \( m \) is:
[tex]\[
m = -2
\][/tex]
2. Identify the y-intercept (b):
The y-intercept (b) is the constant term in the equation, which is the value of \( y \) when \( x = 0 \). In the given equation:
[tex]\[
y = -2x - 2
\][/tex]
The constant term is \( -2 \). Hence, the y-intercept \( b \) is:
[tex]\[
b = -2
\][/tex]
3. Construct the main components of the line:
Now we have both the slope and the intercept:
[tex]\[
\text{Slope } (m) = -2
\][/tex]
[tex]\[
\text{Y-intercept } (b) = -2
\][/tex]
4. Interpretation of the slope and intercept:
- The slope \( -2 \) indicates that for every unit increase in \( x \), \( y \) decreases by 2 units. This shows a line that is descending from left to right.
- The y-intercept \( -2 \) tells us that the line crosses the y-axis at the point \( (0, -2) \).
5. Graphical Representation:
If you were to graph this equation, you would start at the point \( (0, -2) \) and then use the slope to determine other points on the line. From \( (0, -2) \), you would move 1 unit to the right (positive x direction) and 2 units down (negative y direction) to find the next point \( (1, -4) \). Connecting these points would give you a straight line.
### Summary:
- The slope of the given line is \( -2 \).
- The y-intercept of the line is \( -2 \).
So, the line described by the equation [tex]\( y = -2x - 2 \)[/tex] descends with a slope of [tex]\( -2 \)[/tex] and intersects the y-axis at [tex]\( -2 \)[/tex].
