Problem 58 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given point and has the given slope. $$ (5,5), m=5 $$

Step-by-Step Solution

Answer

So, the equation of the line that passes through the given point with the given slope is \( y - 5 = 5(x - 5) \)

Step 1: Understand the Problem

We need to write an equation or expression based on the given information.

Step 2: Translate to Mathematical Form

We convert the verbal description or conditions into mathematical notation.

Step 3: State the Equation

So, the equation of the line that passes through the given point with the given slope is \( y - 5 = 5(x - 5) \)

Key Concepts

Equation of a Line

The equation of a line is a fundamental concept in geometry and algebra, enabling us to clearly define or graph any straight line in a two-dimensional plane. There are multiple forms for representing the equation of a line:

• **Slope-Intercept Form**: Defined by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.
• **Point-Slope Form**: Expressed as \( y - y_1 = m(x - x_1) \), it is especially useful when you know the slope \( m \) and one point \((x_1, y_1)\) on the line.
• **Standard Form**: Written as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants, and \( A \) and \( B \) are not both zero.

The point-slope form is particularly helpful for quickly writing the equation of a line when given a point and a slope, as it helps encapsulate both of these known quantities directly and intuitively. Once you have this form, you can also convert it to other forms based on your needs, such as to slope-intercept form for graphing purposes or into standard form for solving systems of equations.

Slope

The slope of a line is a measure of its steepness and direction, which is a critical component in the equation of a line. It's represented by \( m \) and calculated as the change in the y-coordinates divided by the change in the x-coordinates between two distinct points on a line. Mathematically, it is expressed as: \[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]Here's why understanding slope is vital:

• **Positive Slope**: Indicates that the line rises from left to right.
• **Negative Slope**: Indicates that the line falls from left to right.
• **Zero Slope**: Represents a horizontal line, where there is no vertical change.
• **Undefined Slope**: Associated with vertical lines, where there is no horizontal change.

The concept of slope also helps determine parallel and perpendicular lines. Parallel lines share the same slope, while perpendicular lines have slopes that multiply to \(-1\). In coordinate geometry, the slope guides us to understand how the line can translate into a visually interpretable graph.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines the fields of algebra and geometry using a coordinate system to study geometric figures. By placing geometric shapes such as lines, circles, and curves within a coordinate plane, we can gain quantitative insights about their properties. In the context of lines, coordinate geometry allows us to:

• **Locate Points**: Understanding how to pinpoint where specific points lie on the Cartesian plane using ordered pairs \((x, y)\).
• **Plot Lines**: Based on their equations, draw and interpret lines directly on the graph, which is particularly useful for visually verifying solutions.
• **Measure Distances and Angles**: Calculate distances between points and angles between lines using formulas derived from the coordinates.
• **Investigate Intersections**: Determine where lines and curves meet by solving their simultaneous equations.

Using coordinate geometry, we transform abstract shapes into something measurable and analyzable, facilitating problem solving, especially when dealing in algebra where visual representation is beneficial.