What is the value of \(x\) that satisfies the equation \(2(x+4)=5 x-7 ?\) A. \(-1\) B. \(\frac{1}{3}\) C. \(\frac{11}{3}\) D. 5 E. \(\frac{43}{3}\)

Algebraic expressions are the cornerstone of algebra and they encompass a combination of numbers, variables (like x or y), and arithmetic operations such as addition, subtraction, multiplication, and division. For instance, in the equation 2(x + 4), 2, x, and 4 are the components of an algebraic expression.

When faced with such an expression, one should simplify it. Simplifying may involve distributing a multiplier over a parenthesis as in the step-by-step solution presented, where the 2 is distributed across the (x + 4). This expansion step changes the equation into a more straightforward form: 2x + 8, setting the stage for further manipulation to find the variable's value.

Isolating variables is a critical strategy in solving linear equations. The goal is to rearrange the equation so that the variable of interest is by itself on one side. This often involves moving terms from one side of the equation to the other.

From our example, to isolate x, we first subtract 2x from both sides to eliminate the variable from the left side, resulting in 8 = 3x - 7. Further, by adding 7 to both sides, we're left with 15 = 3x. Every step is a move closer to isolating the variable completely. Once the variable is by itself, we can solve for it simply, which is the crux of converting complex equations into simpler, solvable forms.

Remember, the objective is to perform the same operation on both sides of the equation, maintaining the equality. Thus, the essence of isolating the variable is to reverse the arithmetic operations imposed on it until it stands alone.

Quadratic equations are a level up from linear equations and take the form ax^2 + bx + c = 0, where a, b, and c are constants and x represents the variable. The equation used in our exercise is linear, not quadratic, because it lacks the x^2 term. However, understanding quadratics is important as they are prevalent in higher-level mathematics.

Quadratic equations can be solved using various methods such as factoring, completing the square, using the quadratic formula, or graphing. While these methods are beyond the scope of our linear equation, it's worth mentioning that a sound grasp of solving linear equations builds a strong foundation for tackling the complexities of quadratic equations.

Linear equations can be seen as a subset of quadratic equations, where the coefficient a of the x^2 term is zero. Hence, mastering linear equations paves the way towards understanding their quadratic counterparts.