First, we need to understand the relationship between the two equations in the system by analyzing and graphing them.
### Step-by-Step Solution:
#### Step 1: Rewrite the equations in a comparable form
1. Equation 1: [tex]\( x + 2y = 4 \)[/tex]
- To put this in slope-intercept form ([tex]\( y = mx + b \)[/tex]), solve for [tex]\( y \)[/tex]:
[tex]\[
x + 2y = 4
\][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
2y = -x + 4
\][/tex]
Divide everything by 2:
[tex]\[
y = -\frac{1}{2}x + 2
\][/tex]
2. Equation 2: [tex]\( y = -\frac{1}{2}x + 2 \)[/tex]
- This equation is already in slope-intercept form.
Now, we see that both equations are [tex]\( y = -\frac{1}{2}x + 2 \)[/tex].
#### Step 2: Graph the Equations
Since both equations are the same [tex]\( y = -\frac{1}{2}x + 2 \)[/tex], we graph just one line:
- Slope ([tex]\(m\)[/tex]): [tex]\(-\frac{1}{2}\)[/tex]
- Y-intercept ([tex]\(b\)[/tex]): 2
1. Start at the y-intercept (0, 2).
2. Use the slope to find another point. Since the slope is [tex]\(-\frac{1}{2}\)[/tex], go down 1 unit and right 2 units to find the next point. Repeat this to plot several points, then draw the line through these points.
#### Step 3: Determine the Number of Solutions
- Since both equations represent the same line, they are coincident lines.
- Coincident lines lie exactly on top of each other, which means every point on one line is also a point on the other line.
### Conclusion:
As both lines are coincident, they have infinitely many points of intersection. Therefore, the system has infinitely many solutions.
Thus, the number of solutions is:
[tex]\[
\text{Infinitely many solutions}
\][/tex]
