Solve for $x$:
$-1 \frac{2}{3} x-2=2 \frac{7}{12}$
The given equation is \(-1 \frac{2}{3}x - 2 = 2 \frac{7}{12}\). To solve this linear equation, we first represent the mixed numbers as improper fractions.
The mixed number \(-1 \frac{2}{3}\) can be converted to an improper fraction:
\[-1 \frac{2}{3} = -\left(\frac{1 \times 3 + 2}{3}\right) = -\frac{5}{3}\]
The mixed number \(2 \frac{7}{12}\) can be converted to an improper fraction:
\[2 \frac{7}{12} = \frac{2 \times 12 + 7}{12} = \frac{24 + 7}{12} = \frac{31}{12}\]
Now, the equation becomes:
\[-\frac{5}{3}x - 2 = \frac{31}{12}\]
To isolate the term with \(x\), we add 2 to both sides of the equation:
\[-\frac{5}{3}x = \frac{31}{12} + 2\]
To perform the addition, we express 2 with a denominator of 12:
\[2 = \frac{2 \times 12}{12} = \frac{24}{12}\]
Adding these fractions:
\[-\frac{5}{3}x = \frac{31}{12} + \frac{24}{12} = \frac{55}{12}\]
To solve for \(x\), we multiply both sides of the equation by the reciprocal of the coefficient \(-\frac{5}{3}\), which is \(-\frac{3}{5}\):
\[x = \frac{55}{12} \times \left(-\frac{3}{5}\right)\]
Multiplying the numerators and denominators:
\[x = -\frac{55 \times 3}{12 \times 5}\]
We can simplify this by canceling common factors:
\[x = -\frac{(11 \times 5) \times 3}{(4 \times 3) \times 5} = -\frac{11 \times 5 \times 3}{4 \times 3 \times 5} = -\frac{11}{4}\]
Thus, the final solution is \(x = -\frac{11}{4}\), which can also be written as \(-2 \frac{3}{4}\) or \(-2.75\).
Linear equations like this are used in various fields, such as finance, physics, and engineering, to find unknown quantities. For example, if you are tracking the balance of a project budget that started with an initial cost and is being reduced by a fixed amount per stage, you can model the remaining budget over time using a linear equation to determine exactly how many stages (represented by the variable \(x\)) can be completed before hitting zero or a specific target value. Mastering these steps ensures you can solve for unknown variables in any linear relationship.