Solve the equation:
$-g+2(3+g)=-4(g+1)$
The problem requires solving the linear equation \(-g + 2(3 + g) = -4(g + 1)\) for the variable \(g\). This involves simplifying both sides of the equation by distributing constants into parentheses, combining like terms, and then isolating the variable \(g\).
First, we distribute the constants on both sides of the equation. On the left side, we have \(2(3 + g) = 6 + 2g\). Substituting this into the equation, we get \(-g + 6 + 2g = -4(g + 1)\). On the right side, we distribute \(-4\), giving \(-4(g + 1) = -4g - 4\). Now the equation is \(-g + 6 + 2g = -4g - 4\).
Next, we combine like terms on each side. On the left, we combine \(-g\) and \(2g\) to get \(g + 6\). The equation now stands as \(g + 6 = -4g - 4\). To isolate \(g\), we add \(4g\) to both sides: \(g + 4g + 6 = -4\), which simplifies to \(5g + 6 = -4\). Subtracting 6 from both sides gives \(5g = -10\). Dividing by 5 results in \(g = \frac{-10}{5} = -2\).
Linear equations are fundamental in many real-world scenarios. For example, imagine you are comparing two different payment plans for a service. Plan A might charge a flat fee plus a hourly rate, and Plan B might charge a different base fee plus a different hourly rate. By setting these two algebraic expressions equal to each other, you can solve for the specific amount of time at which the costs of the two plans are exactly the same, helping you make an informed decision based on your usage.