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

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

GPT-4o mini · Answer

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 ( $y = m x + b$ ) into that format.

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

$2 y = 6 x + 2$

Divide everything by 2: $y = 3 x + 1$
$y - 5 x = 2$

Rearranging to slope-intercept form: $y = 5 x + 2$
$y - 4 x + 1 = 0$

Rearranging to get $y$ by itself: $y = 4 x - 1$

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

$y = 5 x - 4$
$y = - 2 x + 3$
$y = 3 x + 1$
$y = 3 x + 1$
$y = 5 x + 2$
$y = 4 x - 1$

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

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