How many solutions does this equation have?
$7 w-(2+w)=2(3 w-1)$
The equation to be analyzed is \(7w - (2 + w) = 2(3w - 1)\). To determine the number of solutions, we first expand and simplify both sides of the equation. This is a linear equation in one variable, \(w\), which can be written in the general form \(Aw + B = Cw + D\). If \(A = C\) and \(B = D\), the equation holds for all values of \(w\), indicating that there are infinitely many solutions. If \(A = C\) and \(B eq D\), there are no solutions. If \(A eq C\), there is exactly one solution.
First, simplify the left side of the equation:
\[7w - (2 + w) = 7w - 2 - w = 6w - 2.\]
Next, simplify the right side of the equation:
\[2(3w - 1) = 2 \cdot 3w - 2 \cdot 1 = 6w - 2.\]
Now, substitute these simplified forms back into the original equation:
\[6w - 2 = 6w - 2.\]
Comparing both sides of the simplified equation \(6w - 2 = 6w - 2\), we see that the coefficients of \(w\) are identical (\(6 = 6\)) and the constant terms are identical (\(-2 = -2\)). Because both sides of the equation are equivalent for any real number \(w\), the equation represents an identity. Thus, every real number is a solution to the equation. Therefore, there are infinitely many solutions.
Equations with infinitely many solutions often arise in real-world scenarios where different methods lead to the same outcome. For example, consider two different membership plans at a fitness center: Plan A might charge a flat fee plus a rate per visit, while Plan B might be structured differently but ends up costing exactly the same for every possible number of visits. If you set up an equation to find where the costs are equal, you might find that the costs are equivalent regardless of the number of visits, meaning the plans are identical in price. Understanding these identities helps in identifying when two seemingly different options offer the exact same result.