Problem 65 - Intermediate Algebra

Question

What is the point-slope form of the equation of a line?

Step-by-Step Solution

Answer

The point-slope form is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Step 1: Introduction to Point-Slope Form

The point-slope form of a line's equation is used when you know a point on the line and the slope of the line. This form is particularly useful because it allows for easy conversion to other forms, such as the slope-intercept form or the standard form.

Step 2: Understanding the Formula

The point-slope form of the equation of a line is given as \( y - y_1 = m(x - x_1) \). In this equation, \((x_1, y_1)\) is a specific point on the line, and \(m\) represents the slope of the line.

Step 3: Identify the Components

To write the equation of a line in point-slope form, identify a specific point on the line, \((x_1, y_1)\), and the slope of the line, \(m\). These will be used as the parameters in the equation.

Step 4: Write the Equation

Insert the known values of \((x_1, y_1)\) and \(m\) into the formula \( y - y_1 = m(x - x_1) \). This gives you the equation of the line in point-slope form.

Key Concepts

Equation of a Line

An equation of a line represents a straight path on a coordinate plane and can be described in various forms. The point-slope form, slope-intercept form, and standard form are all methods to express the equation of a line. Each form has its unique features and uses.

• **Point-slope form** is useful when you know a point on the line and the line's slope.
• **Slope-intercept form** makes it easy to identify the slope and the y-intercept.
• **Standard form** organizes the equation into a neat, generalized arrangement.

Understanding each form helps in solving for a line's characteristics or converting between forms to suit different problem-solving contexts. This flexibility is particularly beneficial when addressing diverse mathematical problems including systems of equations and graphing tasks.

Slope-Intercept Form

The slope-intercept form of the equation of a line is widely used due to its simplicity and ease of interpretation. It is expressed as:\[y = mx + b\]Here, **\(m\)** denotes the slope of the line, and **\(b\)** is the y-intercept, which is where the line crosses the y-axis. This form is straightforward as it directly showcases the slope and y-intercept.

• **Slope** \((m)\): Describes how steep the line is. A larger value indicates a steeper incline.
• **Y-intercept** \((b)\): The point where the line crosses the y-axis. It provides a starting point for graphing the line.

Using the slope-intercept form, one can quickly graph a line by plotting the y-intercept and using the slope to determine the rise over run from that point. This form is immensely helpful in visual analyses and is often the go-to format for linear equations in algebra courses.

Standard Form

The standard form of a linear equation presents the equation as a clean and orderly statement:\[Ax + By = C\]In this formula, \(A\), \(B\), and \(C\) are integers, with \(A\) and \(B\) not both zero. The standard form is particularly useful in several mathematical scenarios such as calculating intersections.

• **Uniformity**: Equations in standard form are often preferred for certain algebraic techniques, like the elimination method in systems of equations.
• **Clarity**: By setting the equation to equal \(C\), you clearly define outcomes and constraints, useful in array and matrix calculations.
• **Adaptability**: Easily converts to other forms such as point-slope or slope-intercept, depending on the problem's needs.

While it may seem more complex initially, mastering standard form enables you to tackle a variety of mathematical challenges with precision and ease.