Question
Graph this line using the slope and \( y \)-intercept \[ y=\frac{3}{10} x+5 \] Click to select points on the graph.
Ask by Montgomery Cole. in the United States
Jan 23,2025
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Answer
To graph the line \( y = \frac{3}{10}x + 5 \), plot the \( y \)-intercept at \( (0, 5) \). Then, use the slope \( \frac{3}{10} \) to find another point by moving 10 units right and 3 units up to \( (10, 8) \). Draw a straight line through these points to complete the graph.
Solution
To graph the line \( y = \frac{3}{10}x + 5 \) using its slope and \( y \)-intercept, follow these steps:
### 1. Identify the \( y \)-Intercept
The equation is in the slope-intercept form \( y = mx + b \), where:
- \( m \) is the slope.
- \( b \) is the \( y \)-intercept.
For the given equation:
- **Slope (\( m \))**: \( \frac{3}{10} \)
- **\( y \)-Intercept (\( b \))**: \( 5 \)
**Plot the \( y \)-intercept:**
- Start by plotting the point \( (0, 5) \) on the Cartesian plane.
### 2. Use the Slope to Find Another Point
The slope \( \frac{3}{10} \) means that for every 10 units you move horizontally (to the right), you move up 3 units vertically.
**Starting from \( (0, 5) \):**
- Move **10 units right** along the \( x \)-axis to \( x = 10 \).
- Move **3 units up** to reach \( y = 8 \).
**Plot the second point:** \( (10, 8) \)
### 3. Draw the Line
- Use a ruler to draw a straight line passing through the points \( (0, 5) \) and \( (10, 8) \).
- Extend the line in both directions.
- Add arrows on both ends to indicate that the line continues infinitely.
### 4. (Optional) Find Additional Points for Accuracy
To ensure accuracy, you can find more points by using the slope or choosing different \( x \)-values.
**Example:**
- Let \( x = -10 \):
\[
y = \frac{3}{10}(-10) + 5 = -3 + 5 = 2
\]
**Point:** \( (-10, 2) \)
- Let \( x = 20 \):
\[
y = \frac{3}{10}(20) + 5 = 6 + 5 = 11
\]
**Point:** \( (20, 11) \)
Plotting these points can help verify the accuracy of your line.
### 5. Visual Representation
Here's a simple representation of the graph on a coordinate plane:
```
y
|
12|
11| * (20,11)
10|
9 |
8 | * (10,8)
7 |
6 |
5 | * (0,5)
4 |
3 |
2 | * (-10,2)
1 |
0 |____________________________ x
-10 -5 0 5 10 15 20
```
- The asterisks (*) represent the plotted points.
- Draw a straight line through these points to complete the graph of the equation.
### Summary
- **Slope (\( m \))**: \( \frac{3}{10} \) (rise over run)
- **\( y \)-Intercept (\( b \))**: \( 5 \) (where the line crosses the \( y \)-axis)
- **Key Points**: \( (0, 5) \), \( (10, 8) \), \( (-10, 2) \), \( (20, 11) \)
By plotting these points and drawing a line through them, you've successfully graphed the equation \( y = \frac{3}{10}x + 5 \).
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Simplify this solution Bonus Knowledge
To graph the line given by the equation \( y=\frac{3}{10} x+5 \), first identify the \( y \)-intercept. Here, the line crosses the \( y \)-axis at \( (0, 5) \). From this point, use the slope \( \frac{3}{10} \), which indicates that for every 10 units you move to the right (increasing \( x \)), you move up 3 units (increasing \( y \)). This gives you another point to plot: \( (10, 8) \). Now draw a straight line through the two points you marked. Voilà, your line is ready! Don’t forget, plotting additional points can confirm the accuracy of your line! Select points such as \( (10, 8) \) and \( (-10, 2) \) to add more structure to your graph.
