If \(12 x=-8(10-x),\) then \(x=?\) \(\begin{array}{l}{\mathbf{F} . \quad 20} \\ {\mathbf{G} .} & {8} \\\ {\mathbf{H} .} & {7 \frac{3}{11}} \\ {\mathbf{J} .} & {6 \frac{2}{13}} \\\ {\mathbf{K} .} & {-20}\end{array}\)

Understanding linear equations is fundamental to mastering algebra. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations appear in the form of \(ax + b = 0\), where \(a\) and \(b\) are constants. These equations are special because they graph a straight line when plotted on a coordinate axis.

The primary goal when solving a linear equation is to find the value of the variable that makes the equation true. This process typically involves combining like terms, which are terms that have the same variable raised to the same power, and then isolating the variable on one side of the equation through addition, subtraction, multiplication, or division. The example \(12x = -8(10 - x)\) is a linear equation where our objective is to find the value of \(x\).

Variables isolation is a technique that aims to get the variable of interest alone on one side of the equation so that the solution becomes obvious. The process begins with simplifying the equation as much as possible by combining like terms and using inverse operations. For instance, if the variable is being added to a number, you would subtract that number from both sides.

Once the equation is simplified, you either add, subtract, divide, or multiply both sides of the equation to get the variable by itself. In our example, after distributing the -8 and simplifying, we moved the variable terms to one side by subtracting \(8x\) from both sides, leading to \(4x = -80\). We then isolated \(x\) by dividing both sides by 4. Isolating the variable is a critical step, as it transforms the equation into \(x =\) some number, which is the most straightforward form of the answer.

The distributive property is an essential mathematical principle that allows you to multiply a single term by each term within a parenthesis in an algebraic expression. It is formally stated as \(a(b + c) = ab + ac\). This property is vital for simplifying equations because it helps eliminate parentheses, which makes it easier to combine like terms and eventually solve for variables.

In the given problem, the distributive property was applied in the first step, where \( -8\) was multiplied by both \(10\) and \( -x\), resulting in the simplified equation \(12x = -80 + 8x\). This step was crucial for converting the equation into a form that allowed us to isolate the variable \(x\) effectively. The proper application of the distributive property often sets the stage for a successful problem-solving process in algebra.