For the first part of the problem, we are given the equation \(5(x-2) = (x-2) + ?(x-2)\). We need to determine the missing term represented by the question mark. Let \(k\) be the missing term. The equation can be rewritten as:
\[5(x-2) = (x-2) + k(x-2)\]
We can factor out \((x-2)\) from the right side of the equation:
\[5(x-2) = (1 + k)(x-2)\]
Assuming \(x
eq 2\), we can divide both sides by \((x-2)\), resulting in:
\[5 = 1 + k\]
Subtracting 1 from both sides gives us:
\[k = 4\]
Thus, the missing term is 4.
For the second part, we are given the equation \(3k + 7 = 8k - 3\) and need to identify the false statement among the provided options. Let's evaluate each option by manipulating the original equation.
Starting with:
\[3k + 7 = 8k - 3\]
Option B: To check \(5k = 10\), we subtract \(3k\) from both sides of the original equation:
\[7 = 5k - 3\]
Then, add 3 to both sides:
\[10 = 5k\]
Since this matches the original equation, statement B is true.
Option C: To check \(3k + 11 = 8k + 1\), we add 4 to both sides of the original equation:
\[3k + 7 + 4 = 8k - 3 + 4\]
\[3k + 11 = 8k + 1\]
Since this matches the original equation, statement C is true.
Option D: To check \(4k + 7 = 9k - 3\), we add \(k\) to both sides of the original equation:
\[3k + k + 7 = 8k + k - 3\]
\[4k + 7 = 9k - 3\]
Since this matches the original equation, statement D is true.
Option A: To check \(3k = 8k + 4\), we start from the original equation \(3k + 7 = 8k - 3\). If we subtract 7 from both sides, we get:
\[3k = 8k - 10\]
Comparing this to the statement in option A (\(3k = 8k + 4\)), we see they are not the same. Therefore, statement A is false.
We have determined that the missing term in the first part is 4 and that statement A is the false statement in the second part. The solutions are concluded.
Understanding how to manipulate algebraic equations and isolate variables is essential in everyday tasks such as balancing a checkbook, calculating travel time, or adjusting recipe portions. For example, if you are scaling a recipe where the amount of flour is \(5 \times (cups - 2)\) and you find a relationship of \((cups - 2) + 4 \times (cups - 2)\), you are essentially applying the same algebraic simplification we used to find the missing term. These fundamental techniques form the basis for solving more complex real-world problems involving relationships between variables.