Question

Which two equations form a system with infinite solutions (same y intercepts)
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 infinite solutions (meaning they are essentially the same line), we need to check if they have the same slope and the same y-intercept.

Let's first rewrite all the given equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

1. \( y = 5x - 4 \) - Slope = 5, y-intercept = -4
2. \( y = -2x + 3 \) - Slope = -2, y-intercept = 3
3. \( y = 3x + 1 \) - Slope = 3, y-intercept = 1
4. \( 2y = 6x + 2 \) → \( y = 3x + 1 \) - Slope = 3, y-intercept = 1
5. \( y - 5x = 2 \) → \( y = 5x + 2 \) - Slope = 5, y-intercept = 2
6. \( y - 4x + 1 = 0 \) → \( y = 4x - 1 \) - Slope = 4, y-intercept = -1

Now summarizing the slopes and y-intercepts:
1. \( y = 5x - 4 \) (Slope = 5, y-intercept = -4)
2. \( y = -2x + 3 \) (Slope = -2, y-intercept = 3)
3. \( y = 3x + 1 \) (Slope = 3, y-intercept = 1)
4. \( y = 3x + 1 \) (Slope = 3, y-intercept = 1)
5. \( y = 5x + 2 \) (Slope = 5, y-intercept = 2)
6. \( y = 4x - 1 \) (Slope = 4, y-intercept = -1)

From this, we can see:
- Equations 3 and 4 are the same: \( y = 3x + 1 \). Thus they have the same slope (3) and the same y-intercept (1).

The pair of equations that form a system with infinite solutions is:
**3 and 4**.