Question

Which two equations form a system with infinite solutions (same slope and y-intercepts) ? ** 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 find the equations that form a system with infinite solutions, we need to identify equations that are equivalent, meaning they have the same slope (m) and y-intercept (b). Let's first convert equations #4, #5, and #6 into the y = mx + b format.

1. **Equation #4: \(2y = 6x + 2\)**

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

2. **Equation #5: \(y - 5x = 2\)**

Add \(5x\) to both sides:
\[
y = 5x + 2
\]

3. **Equation #6: \(y - 4x + 1 = 0\)**

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

Now we have the following equations in y = mx + b format:

1. \(y = 5x - 4\) (from Option 1)
2. \(y = -2x + 3\) (from Option 2)
3. \(y = 3x + 1\) (from option #4 converted)
4. \(y = 5x + 2\) (from option #5 converted)
5. \(y = 4x - 1\) (from option #6 converted)

Next, we need to identify pairs of equations that have the same slope (m) and the same y-intercept (b):

- **Equations 1 and 4:** \(y = 5x - 4\) and \(y = 5x + 2\) (different y-intercepts)
- **Equations 1 and 5:** \(y = 5x - 4\) and \(y = 5x + 2\) (different y-intercepts)
- **Equations 4 and 3:** \(y = 3x + 1\) and \(y = 3x + 1\) (same slope and intercept – infinite solutions)

So the only pair of equations from this list with the same slope and the same y-intercept is:

- Equations 3 and 4: **(y = 3x + 1 and y = 3x + 1)** (both equations yield infinite solutions)

Thus, the answer is the pair of equations in **Option 3 and Option 4.**