Problem 60 - 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. $$ (14,-3), m=\frac{1}{3} $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the given point (14, -3) and has the slope \(1/3\) is \(y = 1/3x - 23/3\).

Step 1: Identify the given Point and Slope

The given point is (14, -3) and the slope, \(m\), is 1/3.

Step 2: Substitute point and slope into the Formula

Now substitute the given point (14, -3) and slope (1/3) into the point-slope form equation: \(y - (-3) = 1/3 (x - 14)\).

Step 3: Simplify the Equation

Simplify the equation by converting \(y - (-3)\) to \(y + 3\) and distribute \(1/3\) on the right to get: \(y + 3 = 1/3x - 14/3\). Lastly, subtract 3 from both sides to isolate \(y\) and get the final equation: \(y = 1/3x - 14/3 - 3 = 1/3x - 23/3\).

Key Concepts

Equation of a Line

The equation of a line is a mathematical expression that defines all the points on a straight line.
It is used in various forms to communicate the properties and characteristics of the line, such as its slope and intercepts. The most common forms of linear equations are:

• Point-Slope Form: \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
• Slope-Intercept Form: \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.
• Standard Form: \( Ax + By = C \), where \(A\), \(B\), and \(C\) are constants.

The key idea is to understand how each form can easily reveal different properties of the line, which can be helpful, depending on the problem you're solving.
For instance, the point-slope form is particularly useful when you know a point on the line and the slope. Conversely, the slope-intercept form is great for quickly graphing the line or determining where it crosses the y-axis.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. It allows us to represent geometric figures in an algebraic manner. This opens up many possibilities for computation and analysis.

• Coordinate System: The most commonly used system is the Cartesian coordinate system, defined by a horizontal x-axis and a vertical y-axis. Each point in this plane can be uniquely identified by an ordered pair (x, y).
• Lines and Slopes: In coordinate geometry, the slope indicates how steep a line is. It is the ratio of the vertical change to the horizontal change between two points on the line.

Understanding coordinate geometry is vital for interpreting the behavior of points and lines. It provides the tools necessary to form equations that describe various shapes and solve mathematical problems involving distance, midpoints, and slopes.
Incorporating algebra into geometry is powerful because it allows you to solve geometric problems in an analytical way, providing a connection between numerical values and spatial figures.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \(m\) represents the slope of the line, and \(b\) is the y-intercept.
This form is incredibly useful for quickly identifying how a line behaves.

• Slope \(m\): This tells us how steep the line is. A positive slope means the line ascends as it moves from left to right, while a negative slope means it descends.
• Y-Intercept \(b\): This is the point where the line crosses the y-axis. Knowing the y-intercept allows you to start graphing the line easily from the y-axis.

The slope-intercept form is typically derived from the point-slope form by simplifying and solving for \(y\). This conversion process helps in graphing the line and understanding intersections.
Students often find working with the slope-intercept form convenient for calculations, especially when dealing with comparisons between different lines, parallelism, and finding slopes from linear equations.