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Isolating variables is a fundamental skill in algebra that involves rearranging an equation to solve for a specific variable—essentially 'isolating' it on one side of the equation. In the example provided, isolating the variable y is essential to rewrite the equation in the slope-intercept form. Beginning with the equation 2x + 7y = 14, subtracting 2x from both sides gives us 7y = -2x + 14. This step moves all terms involving x to one side, thus 'isolating' the terms with y on the other.

It is crucial to perform the same operation on both sides of the equation to maintain the balance, as if you were balancing scales. Remembering that both sides should remain equal is the key to correctly performing algebraic manipulation. After subtracting, we proceed to the next step of dividing each term by the coefficient of y to get the variable alone.

The slope-intercept form of a linear equation is expressed as y = mx + b, where m is the slope of the line and b is the y-intercept. This form provides a quick way to graph a linear equation and understand its properties. After isolating the variable y, the given equation 2x + 7y = 14 is manipulated to fit this form, which reveals the slope -2/7 and the y-intercept 2 in the final equation y = -2/7x + 2.

The slope represents the rate at which y changes with respect to x (rise over run), and the y-intercept is the point where the line crosses the y-axis (when x is equal to zero). Understanding this form makes it easy to graph the line without having to plot numerous points or values.

Algebraic manipulation encompasses various techniques used to solve and simplify equations, including addition, subtraction, multiplication, division, and distribution. The objective is to modify and reorganize equations to make them easier to solve or understand. In our equation, we use algebraic manipulation to solve for a single variable (y).

We start by subtracting 2x from both sides of the original equation to eliminate the x term from the side with y. We then divide all terms by the coefficient of y, which is 7, to get the final form with y by itself. Each step is a deliberate algebraic manipulation aimed at achieving a specific form — in this case, the slope-intercept form y = mx + b. Mastery of algebraic manipulation allows students to tackle a variety of equations and is a foundation for further studies in mathematics.