To determine which graph shows a system of equations with no solution for the equations \( y = 2x + 4 \) and \( -2x + y = 8 \), we can begin by rewriting the second equation in slope-intercept form (i.e., \( y = mx + b \)).
Starting with \( -2x + y = 8 \):
- Add \( 2x \) to both sides: \[ y = 2x + 8 \]
Now we have the following two equations:
- \( y = 2x + 4 \)
- \( y = 2x + 8 \)
Both equations are in slope-intercept form where the slope (m) for both is 2. However, they have different y-intercepts (b). The first equation has a y-intercept of 4, and the second one has a y-intercept of 8.
Since both lines have the same slope but different y-intercepts, they are parallel lines. Parallel lines never intersect, which means the system of equations has no solution.
To identify the correct graph, look for a graph that shows two parallel lines:
- Both lines should rise at the same angle (indicating the same slope).
- The lines should not cross each other at any point.
This representation will correctly depict the situation where the system of equations has no solution.
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