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Isolating a variable is a fundamental technique in solving algebraic equations, involving the rearrangement of terms to get a single variable by itself on one side of the equation. When dealing with equations like \frac{y}{5}-7=-2x\, the goal is to modify the equation so that the variable of interest, in this case \(y\), stands alone.

To do this, operate on both sides of the equation with the same mathematical actions to maintain equality. For \frac{y}{5}-7=-2x\, the steps involve adding 7 to both sides to eliminate the constant on the \(y\)-containing side, leading to \frac{y}{5} = -2x + 7\. Then, to completely isolate \(y\), you need to eliminate the fraction by multiplying both sides by 5, which is the denominator associated with \(y\). This results in \(y = -10x + 35\), where \(y\) is now isolated and clearly defined as a function of \(x\).

Linear equations are the simplest form of algebraic equations that create straight lines when plotted on a graph. They follow the general form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

In the context of the exercise, \(y = -10x + 35\) is a linear equation since \(y\) depends linearly on \(x\). The coefficient of \(x\) indicates the slope of the line, and the constant term represents the y-intercept, where the line crosses the y-axis. Understanding linear equations is a cornerstone of algebra, enabling the solving of problems involving rates of change, trends, and relationships between two variables.

Algebraic manipulation involves the use of arithmetic operations and properties to rearrange and simplify expressions or equations. Mastery in these manipulative skills is essential to solve for unknown variables successfully.

During the process of solving \(\frac{y}{5}-7=-2 x\), both addition and multiplication were employed strategically. Adding 7 to both sides and then multiplying both sides by 5 are examples of algebraic manipulation. Another key aspect is the property of equality, which states that what is done to one side of the equation must also be done to the other side. By following these principles, solving even more complex algebraic equations is achievable.

Function notation is a concise way to represent functions, which are relationships between sets of numbers or objects. The notation involves using symbols like \(f(x)\) or \(g(x)\), where \(f\) or \(g\) represents the function, and \(x\) denotes the variable input.

In the final step of the example, the relationship between \(y\) and \(x\) is expressed as \(y = -10x + 35\). However, using function notation, we could also write this as \(f(x) = -10x + 35\), emphasizing that \(y\) is a function of \(x\). This notation is particularly useful when dealing with multiple functions or when highlighting the dependence of one variable on another.