Sure! Let's break down the questions using the RARE method (Restate, Answer, Reason, and Example).
### Question 1: What does it mean for a linear equation to have infinitely many solutions?
**Restate:** A linear equation having infinitely many solutions means that there are countless values for the variable that satisfy the equation.
**Answer:** This occurs when the equation represents the same line on a graph, leading to an infinite number of points (solutions) along that line.
**Reason:** When you simplify a linear equation and arrive at a statement that is always true, such as "0 = 0" or "x = x," it indicates that the original equation holds for all values of the variable.
**Example:** For instance, in solving the equation `2(4x + 1) = 3x + 5x + 2`, if you simplify both sides and end up with a statement like `0 = 0`, that means every value of `x` is a solution, and thus there are infinitely many solutions.
### Question 2: When solving an equation, how can you tell it will have infinitely many solutions?
**Restate:** You can tell that a linear equation will have infinitely many solutions by looking for specific characteristics in the simplification process.
**Answer:** If, after simplifying the equation, you find that both sides are equivalent and the variables on both sides completely cancel out, it indicates an infinite number of solutions.
**Reason:** This happens because the same expressions are represented differently on both sides of the equation, confirming that the two sides are identical rather than distinct.
**Example:** In the example, simplifying `2(4x + 1) = 3x + 5x + 2` leads to both sides becoming `8x + 2 = 8x + 2`. When you subtract `8x` from both sides, you are left with `2 = 2`, which is always true, indicating infinitely many solutions.
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Feel free to let me know if you need further clarification or assistance on any specific points!