To create an equation with infinite solutions, both sides must be equivalent. This can be achieved by manipulating the left side to match the form of the right side.
We start with the left side: \[ 4x + 10 \]
Now, let’s express the right side in a similar manner with the variable \(x\):
Let’s assume the right side is: \[ x + c \] where \(c\) is a constant that we need to determine.
For the equation to have infinite solutions, we can set the left-hand side equal to a multiple of the right-hand side.
To achieve that, we need to rewrite \(4x + 10\) in terms of \(x\):
Let's express \(4x + 10\) as a function of \(x\): To make them equal, we can choose values such that both sides are equivalent for all \(x\).
Rewrite \(4x + 10\) to resemble \(x + c\):
Let's equate coefficients:
- The coefficient of \(x\) on the left is 4 and on the right is 1.
- This means we can represent the left side as \(4(x) + 10\) and divide by 4 to fit the right side's x-coefficient.
Trying a form: \[ 4x + 10 = 4(x + \frac{10}{4}) \]
This would look like: \[ x + \frac{40}{4} = x + 10 \]
So now we can rewrite: \[ 4x + 10 = 4(x + 2.5) \] or similar proportions.
We can choose:
- \(c = 10\) makes both sides equal:
So, \[ 4x + 10 = x + 10 \]
Thus: The completed equation with infinite solutions is: \[ 4x + 10 = x + 10 \]
So the answer is: \[ \text{(1): } 10 \]
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