Solve for \(g\):
$-9 g-3=-3(3 g+2)$
The goal is to determine the value(s) of \(g\) that satisfy the linear equation \(-9g - 3 = -3(3g + 2)\). We begin by simplifying both sides of the equation to isolate the variable or determine if the equation results in an identity (infinitely many solutions) or a contradiction (no solution).
Expand the right side of the equation by distributing the constant \(-3\):
\[-3(3g + 2) = -3 \cdot 3g + (-3) \cdot 2 = -9g - 6.\]
Now, substitute this expanded form back into the original equation:
\[-9g - 3 = -9g - 6.\]
To solve for \(g\), add \(9g\) to both sides of the equation to group the variable terms:
\[(-9g + 9g) - 3 = (-9g + 9g) - 6\]
\[-3 = -6.\]
Since \(-3 = -6\) is a false statement, the equation has no solution. There is no value of \(g\) that can make the left-hand side equal to the right-hand side. The expression leads to a contradiction.
Imagine you are tracking two different delivery plans for a company. Plan A starts with an initial deficit of 3 units and reduces it at a rate proportional to -9 per unit of time, while Plan B is a scaled version of a different process that results in a deficit of -9 per unit of time minus 6. If you calculate the values and arrive at a contradiction like -3 = -6, it simply means that these two plans are incompatible and can never result in the same outcome, no matter how much time passes. This logic is fundamental in verifying consistency in systems of linear equations.