Overview of Finding the Equation of a Line with a Given Slope and Point

Transforming a mathematical concept from an abstract idea into a tangible equation is a crucial task in algebra and geometry. One such common problem involves finding the equation of a line given its slope and a point it passes through. This article will walk you through the process, using the provided example, to provide a clear and comprehensive guide. We will also ensure that the article is optimized for Google’s search engine standards to help it rank well.

Understanding the Problem

The problem presented states that we need to determine the equation of a line with a slope of (-frac{2}{3}) that passes through the point ((3, 4)). To solve this, we will use the point-slope form of the equation of a line, which is a fundamental concept in coordinate geometry.

Point-Slope Form of a Line

The point-slope form of the equation of a line is given by the formula:

(y - y_1 m(x - x_1))

Where:

(m) is the slope of the line. (x_1, y_1) is the point the line passes through.

Solving the Equation

Given:

(m -frac{2}{3}) Point (P_1(3, 4))

Substituting these values into the point-slope form equation:

(y - 4 -frac{2}{3}(x - 3))

Let's simplify this equation:

(y - 4 -frac{2}{3}x frac{2}{3}3)

Here, we distribute the slope on the right side:

(y - 4 -frac{2}{3}x 2)

Next, we add 4 to both sides to solve for (y):

(y -frac{2}{3}x 2 4)

Combining like terms, we get the equation in slope-intercept form:

(y -frac{2}{3}x 6)

Alternative Formulations and Generalization

The same line can be represented in other forms:

Standard Form

The standard form of a line is (Ax By C). We can write our line in this form as follows:

(3y 2x 18)

General Form

The general form is a combination of the standard and slope-intercept forms, where the coefficients are the same but the variables are arranged differently:

(x 3y - 18 0)

Conclusion and Optimization for SEO

By following the steps and understanding the different forms of line equations, you can efficiently solve similar problems. To optimize this content for SEO, ensure the following:

Use relevant keywords like “equation of a line,” “point-slope form,” and “slope-intercept form.” Incorporate headers and sub-headers that follow a logical structure. Use bullet points for clarity and easy digestion of information. Include mathematical symbols using HTML entities or MathJax for proper rendering.

By adhering to these guidelines, not only will you solve the problem effectively, but you will also make your content readable and SEO-friendly.

Further Resources

For further learning and practice, refer to the following resources:

Math is Fun: Line Equation - Point-Slope Form Khan Academy: Equation of a Line from Its Slope and a Point Math Planet: Equation of a Line