Question

Which two equations form a system with infinite solutions (same y intercepts) ?
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 form a system with infinite solutions, two equations must represent the same line, which means they must be equivalent in form. This occurs when they have the same slope and y-intercept.

Let's analyze the equations you've provided to determine which two are equivalent:

1. **y = 5x - 4**
This has a slope of 5 and a y-intercept of -4.

2. **y = -2x + 3**
This has a slope of -2 and a y-intercept of 3.

3. **y = 3x + 1**
This has a slope of 3 and a y-intercept of 1.

4. **2y = 6x + 2**
Dividing through by 2 gives:
\[
y = 3x + 1
\]
This has a slope of 3 and a y-intercept of 1 (same as equation 3).

5. **y - 5x = 2**
Rearranging gives:
\[
y = 5x + 2
\]
This has a slope of 5 and a y-intercept of 2.

6. **y - 4x + 1 = 0**
Rearranging gives:
\[
y = 4x - 1
\]
This has a slope of 4 and a y-intercept of -1.

After analyzing each equation, we can see:

- Equations **3** and **4** are equivalent because they simplify to \(y = 3x + 1\).
- All other pairs of equations have different slopes or y-intercepts.

Therefore, the two equations that form a system with infinite solutions are **3** and **4**.