Answer :
Let's delve into the step-by-step solution to understand how to graph a line given the components provided.
### Step-by-Step Solution:
1. Identify the Line Equation:
- We are working with the slope-intercept form of a linear equation, which is:
[tex]\[ y = mx + b \][/tex]
- Here, m represents the slope of the line and b represents the y-intercept.
2. Determine the Slope (m) and Y-Intercept (b):
- From the given information, the slope (m) is 3.
- The y-intercept (b) is 6.
3. Equation of the Line:
- Putting the values of m and b into the slope-intercept form, the equation becomes:
[tex]\[ y = 3x + 6 \][/tex]
4. Calculate the Y-coordinate for a Given X-value:
- To graph the line, it's useful to find specific points that lie on the line.
- We are given an x-value of -6 and need to find the corresponding y-value.
- Substitute x = -6 into the equation:
[tex]\[ y = 3(-6) + 6 \][/tex]
- Calculate the result:
[tex]\[ y = -18 + 6 = -12 \][/tex]
- Therefore, when x = -6, y = -12.
5. Plot the Points and Draw the Line:
- We have the y-intercept (0, 6), which is where the line crosses the y-axis.
- We also have the point (-6, -12).
6. Graphing the Line:
- To graph the line, plot the points (0, 6) and (-6, -12) on a coordinate plane.
- Draw a line through these points, extending it in both directions. This is the graph of the equation y = 3x + 6.
### Summary:
- Equation of the Line: y = 3x + 6
- Y-Intercept: (0, 6)
- Point for X-value -6: (-6, -12)
By following these steps, you can graph the line accurately on a Cartesian plane.
### Step-by-Step Solution:
1. Identify the Line Equation:
- We are working with the slope-intercept form of a linear equation, which is:
[tex]\[ y = mx + b \][/tex]
- Here, m represents the slope of the line and b represents the y-intercept.
2. Determine the Slope (m) and Y-Intercept (b):
- From the given information, the slope (m) is 3.
- The y-intercept (b) is 6.
3. Equation of the Line:
- Putting the values of m and b into the slope-intercept form, the equation becomes:
[tex]\[ y = 3x + 6 \][/tex]
4. Calculate the Y-coordinate for a Given X-value:
- To graph the line, it's useful to find specific points that lie on the line.
- We are given an x-value of -6 and need to find the corresponding y-value.
- Substitute x = -6 into the equation:
[tex]\[ y = 3(-6) + 6 \][/tex]
- Calculate the result:
[tex]\[ y = -18 + 6 = -12 \][/tex]
- Therefore, when x = -6, y = -12.
5. Plot the Points and Draw the Line:
- We have the y-intercept (0, 6), which is where the line crosses the y-axis.
- We also have the point (-6, -12).
6. Graphing the Line:
- To graph the line, plot the points (0, 6) and (-6, -12) on a coordinate plane.
- Draw a line through these points, extending it in both directions. This is the graph of the equation y = 3x + 6.
### Summary:
- Equation of the Line: y = 3x + 6
- Y-Intercept: (0, 6)
- Point for X-value -6: (-6, -12)
By following these steps, you can graph the line accurately on a Cartesian plane.
