Question

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.

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**.