Solve for x:
$-8-x=-3(2 x-4)+3 x$
The given equation is \(-8 - x = -3(2x - 4) + 3x\). We want to isolate the variable \(x\) to find its value. The equation involves a linear expression with a distributive term on the right-hand side. Our goal is to expand, simplify, and solve for \(x\).
First, we distribute the \(-3\) across the terms inside the parentheses \((2x - 4)\). Using the distributive property \(a(b + c) = ab + ac\), we get:
\[-3(2x - 4) = -3 \times 2x + (-3) \times (-4) = -6x + 12.\]
Now, substitute this back into the original equation:
\[-8 - x = -6x + 12 + 3x.\]
Next, combine the like terms on the right-hand side of the equation. We have \(-6x\) and \(+3x\):
\[-6x + 3x = -3x.\]
So the equation simplifies to:
\[-8 - x = -3x + 12.\]
Now, move all the terms involving \(x\) to the left-hand side and all constant terms to the right-hand side. Adding \(3x\) to both sides:
\[-8 - x + 3x = 12\]
\[-8 + 2x = 12.\]
Next, add \(8\) to both sides of the equation to isolate the \(x\) term:
\[2x = 12 + 8\]
\[2x = 20.\]
Finally, solve for \(x\) by dividing both sides by the coefficient \(2\):
\[x = \frac{20}{2}\]
\[x = 10.\]
Thus, the solution to the equation is \(x = 10\).
Linear equations are fundamental in everyday scenarios where you need to determine an unknown quantity based on a set of constraints. For example, if you are buying items where you have an initial budget of \(\$20\) but need to account for a discount or a tax structure, you can set up a linear equation to find out how many units you can purchase. Mastering the manipulation of these equations allows you to solve for unknowns in budgeting, planning, and resource allocation problems effectively.