91Ó°ÊÓ

In algebra, the slope-intercept form is a linear equation of the form \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. Understanding this form is crucial because it directly tells you the steepness and direction of the line, as well as where the line meets the y-axis.

To find the slope-intercept form from any linear equation, we aim to isolate the \(y\) variable on one side. This often involves algebraic manipulation to move terms involving \(x\) and constants to the opposite side. Once in this form, the coefficient of \(x\) is immediately identifiable as the slope. If the equation does not have an \(x\) term, it means the slope is zero, indicating a horizontal line.

Algebraic manipulation encompasses various methods used to rearrange and simplify equations. It includes operations such as adding, subtracting, multiplying, or dividing both sides of an equation by the same value. These techniques are essential for isolating variables, finding the slope of a line, or rewriting equations in different forms, such as the slope-intercept form mentioned earlier.

For instance, to isolate \(y\) in the equation given in the original exercise, we subtract \(x\) from both sides and then simplify. Algebraic manipulation is also used to combine like terms, factor expressions, and expand polynomials. Mastery of these techniques is fundamental to your progress in algebra and unlocking the ability to solve various types of equations with ease.

The y-intercept is a specific point on the y-axis where the graph of an equation crosses it. In the slope-intercept form, this is represented by \(b\) in the equation \(y = mx + b\). The y-intercept is key in graphing straight lines because it provides a starting point from which you can then apply the slope to determine the direction of the line.

The y-intercept is particularly important when an equation has no x-term, as in the original exercise where it is determined that the slope is \(0\). In such cases, the equation represents a horizontal line that intersects the y-axis at the y-intercept. For the provided exercise, the y-intercept is \(\frac{2}{3}\), suggesting that the line crosses the y-axis at the point \((0, \frac{2}{3})\).