Question

Which two equations form a system with one solution (different slopes)?
1. y=5×-4
2. y=-2×+3
3. y=3×+1
4. 2y=6×+2
5. y-5×=2
6. y-4×+1=0

Answer 1

To determine which two equations form a system with one solution, we need to identify pairs of equations that have different slopes.

Let's first rewrite each equation in slope-intercept form \(y = mx + b\), where \(m\) is the slope:

1. \(y = 5x - 4\) (slope = 5)
2. \(y = -2x + 3\) (slope = -2)
3. \(y = 3x + 1\) (slope = 3)
4. \(2y = 6x + 2\) → Divide by 2: \(y = 3x + 1\) (slope = 3)
5. \(y - 5x = 2\) → Rearranging: \(y = 5x + 2\) (slope = 5)
6. \(y - 4x + 1 = 0\) → Rearranging: \(y = 4x - 1\) (slope = 4)

Now, let's summarize the slopes of each equation:
1. \(y = 5x - 4\) → slope = 5
2. \(y = -2x + 3\) → slope = -2
3. \(y = 3x + 1\) → slope = 3
4. \(y = 3x + 1\) → slope = 3 (same as equation 3)
5. \(y = 5x + 2\) → slope = 5
6. \(y = 4x - 1\) → slope = 4

To form a system with one solution, we need pairs of equations with different slopes. Here are some pairs:

- \(2\) (slope = -2) and \(6\) (slope = 4)
- \(2\) (slope = -2) and \(3\) (slope = 3)
- \(2\) (slope = -2) and \(4\) (slope = 3)
- \(6\) (slope = 4) and \(3\) (slope = 3)

As one particular example, equations:

- Equation \(2\): \(y = -2x + 3\) (slope = -2)
- Equation \(6\): \(y = 4x - 1\) (slope = 4)

This pair has different slopes and will thus intersect at exactly one point, indicating they form a system with one solution.

Thus, one example of a system with one solution is:

**System:**
- Equation 2: \(y = -2x + 3\)
- Equation 6: \(y = 4x - 1\)