Certainly! To graph the line given by the equation [tex]\( -2x + y = 4 \)[/tex], let's go through the problem step-by-step:
### Step 1: Rearrange the Equation
Start by solving the equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. This will make it easier to plot the line.
[tex]\[
-2x + y = 4
\][/tex]
Add [tex]\( 2x \)[/tex] to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 2x + 4
\][/tex]
### Step 2: Identify Key Points
Let's identify key points that lie on this line. Selecting a few values for [tex]\( x \)[/tex] and calculating the corresponding [tex]\( y \)[/tex]-values will help us.
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 2(0) + 4 = 4
\][/tex]
Point: [tex]\( (0, 4) \)[/tex]
2. When [tex]\( x = -2 \)[/tex]:
[tex]\[
y = 2(-2) + 4 = -4 + 4 = 0
\][/tex]
Point: [tex]\( (-2, 0) \)[/tex]
3. When [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 2(3) + 4 = 6 + 4 = 10
\][/tex]
Point: [tex]\( (3, 10) \)[/tex]
### Step 3: Plot the Points
On a coordinate system:
- Point 1: [tex]\( (0, 4) \)[/tex]
- Point 2: [tex]\( (-2, 0) \)[/tex]
- Point 3: [tex]\( (3, 10) \)[/tex]
### Step 4: Draw the Line
Connect the points with a straight line. Since the equation is linear, the line will continue infinitely in both directions.
### Step 5: Label the Axes and Line
- Label the x-axis as [tex]\( x \)[/tex].
- Label the y-axis as [tex]\( y \)[/tex].
- Optionally, you can write the equation [tex]\( y = 2x + 4 \)[/tex] near the line for reference.
Here's a plotted line based on the information above:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
0 & 4 \\
-2 & 0 \\
3 & 10 \\
\end{array}
\][/tex]
Graphically:
- The line passes through (0, 4), (-2, 0), and (3, 10).
- It has a positive slope, indicating it rises as [tex]\( x \)[/tex] increases.
### Graph Summary
Here is how the graph will be structured:
- The y-intercept is 4 (the point where the line crosses the y-axis when [tex]\( x = 0 \)[/tex]).
- The slope is 2, indicating that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
This thorough approach ensures that we can clearly visualize and understand the graph of the line [tex]\( -2x + y = 4 \)[/tex].
