How many solutions does this equation have?
$4(2 a+1)=8 a+1$
The given equation is \(4(2a + 1) = 8a + 1\). To find the number of solutions, we need to simplify the equation and determine if it leads to a valid value for \(a\), an identity, or a contradiction.
First, we apply the distributive property to the left-hand side of the equation:
\[4 \times (2a + 1) = 4 \times 2a + 4 \times 1 = 8a + 4.\]
So the equation becomes:
\[8a + 4 = 8a + 1.\]
Next, we attempt to isolate the variable \(a\). Subtracting \(8a\) from both sides of the equation, we get:
\[(8a + 4) - 8a = (8a + 1) - 8a,\]
which simplifies to:
\[4 = 1.\]
This statement, \(4 = 1\), is a contradiction. It is mathematically impossible for \(4\) to equal \(1\). Because the simplification of the original equation leads to a false statement regardless of the value of \(a\), there is no value of \(a\) that can satisfy the original equation.
Since the equation leads to a contradiction, we conclude that there are no values of \(a\) that satisfy the equality. Therefore, the equation has no solutions. Looking at the given options (a. no solution, b. one solution, c. two solutions, d. infinitely many solutions), the correct choice is (a). The final answer is \(\boxed{\text{a}}\).
Imagine you are trying to calculate the price of a certain item where the total cost is determined by two different store policies. If Policy A calculates the cost as 4 times the sum of twice the item's unit price plus one, and Policy B calculates it as 8 times the unit price plus one, you are solving for a price that makes both policies equal. If your calculations lead to an impossible statement like \(4 = 1\), it simply means that there is no unit price where both store policies charge the same amount. This is a common way to analyze if a system of constraints has any valid solution.