Question

Which of the graphs with no solution correctly shows the system of equations y=3x+4 and −3x+y=8 ?

Answer 1

To determine which graph correctly shows the system of equations \( y = 3x + 4 \) and \( -3x + y = 8 \) having no solution, we first need to analyze the equations.

1. **First equation:** \( y = 3x + 4 \)
- This is in slope-intercept form (y = mx + b), where the slope (m) is 3 and the y-intercept (b) is 4. The line will rise steeply.

2. **Second equation:** \( -3x + y = 8 \)
- To put this in slope-intercept form, we can solve for \( y \):
\[
y = 3x + 8
\]
- Here, the slope is also 3, but the y-intercept is 8.

**Comparison of the lines:**
- Both lines have the same slope (3), which means they are parallel.
- The y-intercepts are different (4 and 8), indicating that the lines will never meet.

Since the two lines are parallel and never intersect, they represent a system of equations with no solution.

**Graph characteristics for no solution:**
- The graph should show two parallel lines, one with a y-intercept of 4 and the other with a y-intercept of 8.

Look for a graph that accurately depicts these characteristics to identify the correct representation of the system with no solution.