91Ó°ÊÓ

The point-slope form is a way of expressing the equation of a line using a known point and the slope of the line. The general format is \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope, and \((x_1, y_1)\) is a specific point that lies on the line. This form is particularly useful when you know the slope and one point on the line, and it allows for a straightforward substitution of these values to create a specific line equation.

To understand this clearly, imagine you have a hill (the line), and you know how steep that hill is (the slope \( m \)) and a particular beacon along the path (the point \((x_1, y_1)\)). By using the point-slope form, you can describe the entire hill knowing these two pieces of information.

In our exercise, by substituting \( m = \frac{2}{3} \) and the point \((-8, 9)\) into the point-slope formula, we obtained the equation:

• \( y - 9 = \frac{2}{3}(x + 8) \)

This is the initial form of the line equation we can use for further manipulations.

The standard form of a linear equation is expressed as \( Ax + By = C \). This form is often preferred because it clearly shows the relationship between the x and y variables on the same side and the constant on the other.

Rearranging to this form from point-slope or slope-intercept forms involves algebraic manipulations such as eliminating fractions and grouping terms appropriately. It's essential to note that in the standard form, \( A \), \( B \), and \( C \) are usually integers, and \( A \) should be a non-negative integer for the conventional standard form.

In our task, after expanding and simplifying the point-slope equation, we reached the standard form with the steps:

• Eliminating the fraction by multiplying through by 3: \( 3(y - 9) = 2(x + 8) \)
• Expanding to \( 3y - 27 = 2x + 16 \)
• Rearranging to gather \( x \) and \( y \) terms on one side: \( -2x + 3y = 43 \)

This is the standard form \( -2x + 3y = 43 \), ensuring a positive x coefficient by multiplying by -1, simplifying to \( 2x - 3y = -43 \).

Algebraic manipulations involve rearranging and simplifying equations to achieve a desired form. With line equations, this can mean moving variables around, eliminating fractions, or ensuring coefficients align with a specific standard.

To do this effectively:

• Identify fractions that need clearing by multiplying each term by the denominators' least common multiple.
• Distribute any factors across terms thoroughly to expand equations as needed.
• Like terms are combined, and equation terms are shifted between sides until the required equation form, such as the standard form, is achieved.

In our exercise, these maneuvers started by multiplying the point-slope form to eliminate fractions, soon leading to rearrangement for the standard form. Even further, we simplified by changing all coefficients to integers and ensuring \( A \) (the \( x \)-coefficient) stayed positive in the final standard equation form. This is the beauty and precision of algebraic manipulation at work!