Identify any two equations that form a system with one solution (different slopes) ? [ Hint: Make sure all equations are in Y=mx + b format first. ]

Question

1. y=5x-4
2. y=-2x+3
3. y=3x+1
4. 2y=6x+2
5. y-5x=2
6. y-4x+1=0

Answer 1

To identify two equations that form a system with one solution (which means they have different slopes), we'll convert any equations that aren't already in the \( y = mx + b \) format.

Let's rewrite each equation in the \( y = mx + b \) format when necessary:

1. \( y = 5x - 4 \) (already in the correct format)

2. \( y = -2x + 3 \) (already in the correct format)

3. \( y = 3x + 1 \) (already in the correct format)

4. \( 2y = 6x + 2 \)

   Dividing everything by 2: \[ y = 3x + 1 \] (This is equivalent to equation 3 and does not form a different slope.)

5. \( y - 5x = 2 \)

   Rearranging gives: \[ y = 5x + 2 \]

6. \( y - 4x + 1 = 0 \)

   Rearranging gives: \[ y = 4x - 1 \]

Now let's summarize the equations:

1. \( y = 5x - 4 \) (slope = 5)
2. \( y = -2x + 3 \) (slope = -2)
3. \( y = 3x + 1 \) (slope = 3)
4. \( y = 3x + 1 \) (slope = 3, equivalent to equation 3)
5. \( y = 5x + 2 \) (slope = 5)
6. \( y = 4x - 1 \) (slope = 4)

Now we can see the slopes clearly:

• Equation 1: slope = 5
• Equation 2: slope = -2
• Equation 3: slope = 3
• Equation 5: slope = 5
• Equation 6: slope = 4

To form a system with one solution, we must choose equations with different slopes.

Possible pairs with different slopes:

• Equations 1 and 2:

  • \( y = 5x - 4 \) and \( y = -2x + 3 \)

• Equations 2 and 6:

  • \( y = -2x + 3 \) and \( y = 4x - 1 \)

• Equations 2 and 3:

  • \( y = -2x + 3 \) and \( y = 3x + 1 \)

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

1. \( y = 5x - 4 \)
2. \( y = -2x + 3 \)

These two equations have different slopes and will intersect at one point (one solution).