Solve the literal equation for $x$.
$3 x=6 y-12$
We are given the literal equation \(3x = 6y - 12\) and need to solve for \(x\). This means we want to isolate \(x\) on one side of the equation, leaving only an expression involving \(y\) and constants on the other side. The variables are \(x\) and \(y\), and the coefficients are constants. We will proceed by using basic algebraic operations to manipulate the equation.
To isolate \(x\), we divide both sides of the equation by the coefficient of \(x\), which is \(3\). Applying this to the entire right side, we have:
\[x = \frac{6y - 12}{3}\]
Using the property of division over addition/subtraction, we distribute the division by 3 to both terms on the right side:
\[x = \frac{6y}{3} - \frac{12}{3}\]
Simplifying the fractions, we get:
\[x = 2y - 4\]
This is the expression for \(x\) in terms of \(y\). Comparing this result to the given options, it corresponds to the third option.
Literal equations are used extensively in physics and engineering to rearrange formulas to solve for a specific variable. For example, if you have the formula for the perimeter of a rectangle \(P = 2l + 2w\), you might need to solve for the width \(w\) to find how much material is needed for a specific length \(l\). Understanding how to manipulate these equations is essential for solving real-world problems where variables depend on each other.