91Ó°ÊÓ

Isolating variables is a fundamental skill in algebra, particularly when it comes to writing linear equations. This process involves rearranging an equation so that one specific variable stands alone on one side of the equation. In the case of linear equations, we often aim to isolate the dependent variable, usually represented by \(y\), to express it in terms of the independent variable, typically \(x\).

To do this, one must perform operations that 'undo' what is being done to the variable we want to isolate. For example, if the variable is being added to another term, we subtract that term from both sides. If the variable is being multiplied by a number, we divide both sides by that number, maintaining the balance of the equation.

In the given exercise, we isolated \(y\) from the equation \(2x + 3y = 9\) by first subtracting \(2x\) from both sides, and then dividing each term by 3. This step-by-step approach ensures \(y\) is by itself on one side of the equal sign, essentially 'isolating' it.

Simplifying equations means reducing them to their simplest form, making them easier to interpret or solve. Simplification often involves combining like terms, distributing factors, and reducing fractions to their lowest terms.

It's important to remember that while simplifying, the equation's balance must be preserved. That means whatever operation you perform on one side of the equation, you must do equally to the other side.

In our working example, after isolating \(y\), we simplified the equation by dividing all terms by 3. This turned \(3y = -2x + 9\) into \(y = -\frac{2}{3}x + 3\). The equation is now simpler because it has fewer terms, and each term is as reduced as possible. The resulting equation is easier to use, especially when graphing or doing further algebraic manipulations.

The slope-intercept form is a specific way to write linear equations. It's given by the formula \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept. The slope indicates how steep the line is, and the y-intercept signifies where the line crosses the y-axis.

This form is particularly useful because it allows us to easily graph the linear equation and understand the rate at which the dependent variable changes with the independent variable.

For instance, in the exercise's final solution, the equation \(y = -\frac{2}{3}x + 3\) tells us that the line slopes downward to the right (since the slope is negative) and crosses the y-axis at \(y = 3\). Understanding the slope-intercept form is key to analyzing and visualizing linear relationships in a clear and concise manner.