Problem 27 - Algebra A Combined Function

Question

Find an equation of each line with the given slope that passes through the given point. Write the equation in the form $A x+B y=C. $$ m=\frac{3}{2} ; \quad(5,-6) $$

Step-by-Step Solution

Answer

The equation of the line is \(3x - 2y = 27\).

Step 1: Understand the Point-Slope Form

The point-slope form of a line is given by the equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope of the line. In this problem, \(m = \frac{3}{2}\) and the point is \((5, -6)\).

Step 2: Substitute the Slope and Point into the Point-Slope Form

Substitute \(m = \frac{3}{2}\), \(x_1 = 5\), and \(y_1 = -6\) into the point-slope equation. This becomes \(y + 6 = \frac{3}{2}(x - 5)\).

Step 3: Simplify the Equation

First, distribute \(\frac{3}{2}\) into \(x - 5\). This results in \(y + 6 = \frac{3}{2}x - \frac{15}{2}\).

Step 4: Move to Standard Form

To convert the equation into \(Ax + By = C\) form, move all terms to one side. First, subtract \(y\) and \(6\) from both sides: \(0 = \frac{3}{2}x - y - \frac{15}{2} - 6\).

Step 5: Multiply to Eliminate Fractions

To clear the fractions, multiply the entire equation by 2: \(0 = 3x - 2y - 15 - 12\).

Step 6: Simplify to Standard Form

Simplify the equation: \(3x - 2y = 27\). This is the standard form of the line \(Ax + By = C\).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is an incredibly handy tool when you're given a single point and the slope of a line. It is expressed as \(y - y_1 = m(x - x_1)\). Think of \((x_1, y_1)\) as your starting point or anchor on the line, and \(m\) (the slope) as a measure of the line's steepness or incline.

For example, if you're given the slope \(m = \frac{3}{2}\) and a point \((5, -6)\), you simply plug these values into the formula. This gives you the equation \(y + 6 = \frac{3}{2}(x - 5)\).

The point-slope form is especially helpful because:

• It gives you a direct way to write the equation of a line if you know the slope and a point.
• It's a great starting point for converting to other forms, like the standard form.

You can say the point-slope form acts as a bridge from knowing basic line properties to deriving complete line equations.

Slope

The slope of a line is a key concept in understanding linear equations. Denoted as \(m\), the slope indicates how much the y-value of a point on the line changes for a unit increase in the x-value. In simple terms, it's a measure of how steep a line is.

A positive slope, like \(\frac{3}{2}\), means the line rises as you move from left to right, while a negative slope indicates a downward trend. When the slope is zero, the line is perfectly horizontal, showing no vertical change as you move along it.

The slope is calculated by the formula:

• For two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is \(\frac{y_2 - y_1}{x_2 - x_1}\).

Understanding the slope helps you grasp the behavior of all linear functions and is pivotal when using the point-slope form to write linear equations.

Linear Equation Transformation

Transforming a linear equation from one form to another, such as from point-slope to standard form, involves algebraic maneuvering that solidifies understanding of how equations describe lines.

Starting from the point-slope form, \(y + 6 = \frac{3}{2}(x - 5)\), we aim to achieve the standard form \(Ax + By = C\). The transformation process is as follows:

First, distribute the slope \(\frac{3}{2}\) through \(x - 5\), leading to \(y + 6 = \frac{3}{2}x - \frac{15}{2}\). Adjust terms to move y and constant to one side as \(0 = \frac{3}{2}x - y - \frac{15}{2} - 6\).

To eliminate fractions and simplify conditions, multiply the equation by 2. This yields \(3x - 2y = 27\), a cleaner look at the relationship between x and y.

Why these transformations matter:

• Standard form is particularly favored for easily identifying intercepts and analyzing equations in broader contexts.
• Being able to switch forms allows flexibility in mathematical applications, catering to different problem-solving needs.

Mastery of transformations is a key skill in algebra, encapsulating knowledge of linear relationships in diverse and adaptable ways.