Sure, to graph the equation [tex]\( y = -5x - 3 \)[/tex], we need to follow a series of steps. Let's go through the process in a detailed manner:
### Step-by-Step Solution
1. Identify the equation:
The equation given is [tex]\( y = -5x - 3 \)[/tex]. This is a linear equation in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- The slope ([tex]\( m \)[/tex]) is -5.
- The y-intercept ([tex]\( b \)[/tex]) is -3.
2. Plot the y-intercept:
Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis.
- At [tex]\( x = 0 \)[/tex], [tex]\( y = -3 \)[/tex].
So, the point (0, -3) is the y-intercept.
3. Use the slope to find another point:
The slope of -5 means that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 5 units. We can use this to find another point on the line.
- Starting from the y-intercept (0, -3), if [tex]\( x = 1 \)[/tex], then [tex]\( y = -5(1) - 3 = -8 \)[/tex].
So, the point (1, -8) is another point on the line.
4. Draw the line:
Using the points (0, -3) and (1, -8), draw a straight line through them, extending the line in both directions and adding arrows at the ends to indicate that the line continues infinitely.
5. Check additional points:
We can now calculate a few more points to ensure the accuracy of our graph. Here are some calculated points:
- [tex]\( x = -10 \)[/tex]: [tex]\( y = 47 \)[/tex]
- [tex]\( x = 5 \)[/tex]: [tex]\( y = -28 \)[/tex]
- [tex]\( x = -5 \)[/tex]: [tex]\( y = 22 \)[/tex]
- [tex]\( x = 10 \)[/tex]: [tex]\( y = -53 \)[/tex]
### Final Graph
When we plot these points and connect them with a straight line, we get the graph of the equation [tex]\( y = -5x - 3 \)[/tex]. The line will have a steep negative slope, indicating that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases rapidly.
Make sure to label your axes and points where significant intersections happen. The key points mentioned should be marked as follows for clarity:
- (0, -3)
- (1, -8)
- Any additional points like (2, -13) or (-2, 7)
By following these steps, we ensure that the line accurately represents the equation [tex]\( y = -5x - 3 \)[/tex] on a graph.
