Which two equations form a system with no solutions (same slope but different y intercept)

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

1 answer

To determine which two equations form a system with no solutions, we need to identify equations that have the same slope but different y-intercepts. This will happen when the lines represented by the equations are parallel.

Let's rewrite each equation in slope-intercept form (y = mx + b) to easily identify the slope (m) and y-intercept (b).

  1. \( y = 5x - 4 \)

    • Slope: \( m = 5 \)
    • Y-intercept: \( b = -4 \)
  2. \( y = -2x + 3 \)

    • Slope: \( m = -2 \)
    • Y-intercept: \( b = 3 \)
  3. \( y = 3x + 1 \)

    • Slope: \( m = 3 \)
    • Y-intercept: \( b = 1 \)
  4. \( 2y = 6x + 2 \)

    • Divide by 2:
    • \( y = 3x + 1 \)
    • Slope: \( m = 3 \)
    • Y-intercept: \( b = 1 \)
  5. \( y - 5x = 2 \)

    • Rearranging gives:
    • \( y = 5x + 2 \)
    • Slope: \( m = 5 \)
    • Y-intercept: \( b = 2 \)
  6. \( y - 4x + 1 = 0 \)

    • Rearranging gives:
    • \( y = 4x - 1 \)
    • Slope: \( m = 4 \)
    • Y-intercept: \( b = -1 \)

Now let's compare the slopes:

  • \( y = 5x - 4 \) and \( y = 5x + 2 \) both have a slope of 5 (same slope) but different y-intercepts (-4 and 2). Therefore, they are parallel lines and form a system with no solutions.

So, the two equations that form a system with no solutions are:

  1. \( y = 5x - 4 \) (Equation 1)
  2. \( y = 5x + 2 \) (Equation 5)