Which two equations form a system with no solutions (same slope but different y intercept)? **Hint: You must write # 4 - #6 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

Option 1
Option 2
Option 3
Option 4
Option 5
Option 6

Answer 1

To determine which two equations from the provided list form a system with no solutions, we need to convert any equations that are not already in slope-intercept form (\(y = mx + b\)) into that format.

Let’s convert equations #4, #5, and #6 into slope-intercept form:

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

   Divide everything by 2: \[ y = 3x + 1 \]

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

   Rearranging to slope-intercept form: \[ y = 5x + 2 \]

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

   Rearranging to get \(y\) by itself: \[ y = 4x - 1 \]

Now we have the following equations in slope-intercept form:

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

Next, we need to identify pairs of equations that have the same slope (m) but different y-intercepts (b).

1. Equation 1: Slope = 5, Intercept = -4
2. Equation 2: Slope = -2, Intercept = 3
3. Equation 3: Slope = 3, Intercept = 1
4. Equation 4: (Identical to #3) Slope = 3, Intercept = 1
5. Equation 5: Slope = 5, Intercept = 2
6. Equation 6: Slope = 4, Intercept = -1

From the slopes:

• Equations 1 and 5 both have a slope of 5 but different intercepts (-4 and 2).
• Equations 3 and 4 are identical (same slope and intercept).
• Equations 2 and 6 do not have the same slope as any other equation.

Thus, the pairs of equations with the same slope but different y-intercepts, leading to no solutions, are:

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

Therefore, the answer is: Option 1 and Option 5.