How many solutions will the following equation have?
$2(2 m+6)-6=4 m$
The given equation is \(2(2m + 6) - 6 = 4m\). We are asked to determine the number of solutions for this equation. To solve this, we first simplify the expression on the left-hand side. We start by distributing the constant \(2\) across the terms inside the parentheses. This gives us:
\[2 \times (2m) + 2 \times 6 - 6 = 4m\]
\[4m + 12 - 6 = 4m\]
Now, we simplify the constants by subtracting \(6\) from \(12\), which yields:
\[4m + 6 = 4m\]
Next, we aim to isolate the variable \(m\). We can subtract \(4m\) from both sides of the equation to group all terms involving \(m\) on one side:
\[4m - 4m + 6 = 4m - 4m\]
\[6 = 0\]
This results in a statement \(6 = 0\), which is a mathematical contradiction. Because this statement is never true for any value of \(m\), it implies that there are no values of \(m\) that satisfy the original equation. Therefore, the equation has no solution.
Imagine you have two different cell phone plans. Plan A costs a base fee plus a rate per minute that simplifies to the same total as Plan B in a way that, after accounting for all fees, leads to an impossible equality like \(6 = 0\). This would indicate that the two plans can never cost the same amount of money, regardless of how many minutes you use. Understanding when equations have no solution helps in analyzing such real-world trade-offs and decision-making scenarios.