To graph the line represented by the equation [tex]\( -3x + y = 3 \)[/tex], follow these detailed steps:
1. Rearrange the Equation:
Start by solving the equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
[tex]\[
-3x + y = 3
\][/tex]
Add [tex]\( 3x \)[/tex] to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 3 + 3x
\][/tex]
2. Identify Key Points:
To graph the line, it’s helpful to identify some key points that satisfy the equation. To do this, select a few values for [tex]\( x \)[/tex] and compute the corresponding [tex]\( y \)[/tex] values.
- When [tex]\( x = -2 \)[/tex]:
[tex]\[
y = 3 + 3(-2) = 3 - 6 = -3
\][/tex]
Point: [tex]\((-2, -3)\)[/tex]
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 3 + 3(0) = 3
\][/tex]
Point: [tex]\((0, 3)\)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 3 + 3(1) = 3 + 3 = 6
\][/tex]
Point: [tex]\((1, 6)\)[/tex]
3. Plot the Points:
Plot these points on a coordinate plane:
- [tex]\((-2, -3)\)[/tex]
- [tex]\((0, 3)\)[/tex]
- [tex]\((1, 6)\)[/tex]
4. Draw the Line:
Use a ruler to draw a straight line through these points. This line represents all the solutions to the equation [tex]\( y = 3 + 3x \)[/tex].
5. Verify Slope and Intercepts:
- Slope: The coefficient of [tex]\( x \)[/tex] is 3, so the line has a slope of 3. This means for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 3 units.
- Y-intercept: The line crosses the [tex]\( y \)[/tex]-axis at [tex]\( (0, 3) \)[/tex].
To summarize the information on the graph:
- The equation of the line is [tex]\( y = 3 + 3x \)[/tex].
- The slope of the line is 3.
- The y-intercept is 3.
- Points like [tex]\((-2, -3)\)[/tex], [tex]\((0, 3)\)[/tex], and [tex]\((1, 6)\)[/tex] lie on this line.
### Example of the Graph:
```
10 |
|
|
5 | *
|
|
|---------------------------
|
|
-5 |
|
|
|
-10 |
-10 -5 0 5 10
```
Hence, the graph of the line [tex]\( -3x + y = 3 \)[/tex] is a straight line with the appropriate points plotted and the line passing through these points.
