To graph the linear equation [tex]\(3x + y = 6\)[/tex], follow these steps:
1. Rewrite the Equation in Slope-Intercept Form:
The given equation is [tex]\(3x + y = 6\)[/tex]. We'll solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[
y = -3x + 6
\][/tex]
This is now in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
2. Identify the Slope and Y-Intercept:
- Slope ([tex]\(m\)[/tex]): [tex]\(-3\)[/tex]
- Y-Intercept ([tex]\(b\)[/tex]): [tex]\(6\)[/tex]
3. Plot the Y-Intercept:
- Start by plotting the y-intercept [tex]\((0, 6)\)[/tex] on the graph. This is where the line crosses the y-axis.
4. Use the Slope to Find Another Point:
- The slope is [tex]\(-3\)[/tex], which means for every 1 unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by 3 units. Starting from the y-intercept [tex]\((0, 6)\)[/tex]:
- Move 1 unit to the right (increase [tex]\(x\)[/tex] by 1): [tex]\(0 + 1 = 1\)[/tex]
- Move 3 units down (decrease [tex]\(y\)[/tex] by 3): [tex]\(6 - 3 = 3\)[/tex]
- This gives us the next point [tex]\((1, 3)\)[/tex].
5. Plot the Second Point:
- Now plot the point [tex]\((1, 3)\)[/tex] on the graph.
6. Draw the Line:
- Use a ruler or a straight edge to draw a line through the points [tex]\((0, 6)\)[/tex] and [tex]\((1, 3)\)[/tex],
- Extend the line in both directions.
To verify the accuracy of your graph, you may want to find and plot one or more additional points. For example, solve for [tex]\(x\)[/tex] when [tex]\(y = 0\)[/tex]:
[tex]\[
3x + 0 = 6 \implies x = 2
\][/tex]
So, plot [tex]\((2, 0)\)[/tex] and ensure it lies on the line drawn. In doing so, we verify that our line is correctly positioned according to [tex]\(3x + y = 6\)[/tex].
By following these steps, you should obtain a precise graph for the linear equation [tex]\(3x + y = 6\)[/tex].
