To create an equation that has infinitely many solutions, both sides of the equation must be equivalent. This means that if you manipulate the left side, you should be able to express the right side in the same form.
Let's start with the left side of your equation:
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Given: \( 4x + 10 \)
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We need to find a form for the right side that will make the equation equivalent. A simple way is to take the left side and express it in terms of a single variable:
Suppose we set the right side as: \[ ax + b \]
Now, for the equation to hold true for all \( x \), we need: \[ ax + b = 4x + 10 \]
So we can set \( a = 4 \) and \( b = 10 \) to maintain equality. But to allow for infinitely many solutions, we can express it in a way where both \( a \) and \( b \) can vary while keeping the same ratio.
For example, we could have: \[ 4x + 10 = 4x + 10 \]
Alternatively, you could multiply the entire right side by a non-zero constant \( k \): \[ 4x + 10 = k(4x + 10) \text{ (for any non-zero } k\text{)} \]
Thus, one such solution could be: \[ 4x + 10 = 4x + 10 \]
or more generally:
\[ 4x + 10 = 2(2x + 5) \]
In terms of your request for specific numbers to fill the blanks:
- Let’s fill it as \( ax + b = yx + z \):
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Example I: Fill in the blanks with the same numbers: \[ 4x + 10 = 4x + 10 \]
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Example II: Or a different equivalent form: \[ 4x + 10 = 2(2x + 5) \text{ which simplifies to } 4x + 10. \]
So the complete equation could be: \[ 4x + 10 = 4x + 10 \] or \[ 4x + 10 = 2(2x + 5). \]
Both yield infinite solutions!
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