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:
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y = 5x - 4
This has a slope of 5 and a y-intercept of -4. -
y = -2x + 3
This has a slope of -2 and a y-intercept of 3. -
y = 3x + 1
This has a slope of 3 and a y-intercept of 1. -
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). -
y - 5x = 2
Rearranging gives: \[ y = 5x + 2 \]
This has a slope of 5 and a y-intercept of 2. -
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.