Problem 32 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given point and has the given slope. $$ (-10,0), m=2 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y = 2(x+10)\).

Step 1: Identify the Given Point and Slope

The given point through which the line passes is (-10,0) and the given slope of the line is 2.

Step 2: Write Down the Point-Slope Form

The point slope format is \(y - y_1 = m(x - x_1)\). In this case, \(x_1 = -10, y_1 = 0\), and \(m = 2\).

Step 3: Substitute the Values into the Equation

Substitute -10 for \(x_1\), 0 for \(y_1\), and 2 for \(m\) in the formula. This gives \(y - 0 = 2(x - -10)\). Solve the equation to simplify it.

Step 4: Simplify the Equation

Solving the equation, we get \(y = 2(x+10)\). This is the equation of the line in point-slope form.

Key Concepts

Equation of a Line

An equation of a line is a mathematical description that represents all points lying on that line. It is used to understand the relationship between the coordinates of points on the line. One way to express this equation is through the point-slope form, which conveniently uses a known point and the line's slope. There are several forms of the equation of a line, such as:

• 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 integers.
• Point-slope form: \( y - y_1 = m(x - x_1) \), using a specific point \((x_1, y_1)\) and slope \( m \).

We use the point-slope form when a point on the line and the slope are known, allowing an easy setup of the line's equation. This form is particularly helpful in quickly writing the equation when presented with the slope and point, for instance, \( (-10, 0) \) and \( m = 2 \). Here, plugging into \( y - y_1 = m(x - x_1) \) allows immediate formation of the line's equation.

Slope

The slope of a line is a measure of its steepness, direction, and rate of change. Mathematically, it is the ratio of the vertical change to the horizontal change between two points on the line. In simpler terms, slope tells us how much \( y \) changes for every unit change in \( x \). If you picture a hill, the slope describes how steeply the hill rises or falls.
The formula for finding slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line. A positive slope indicates the line moves upwards from left to right, while a negative slope means it moves downwards. A zero slope is a horizontal line, and an undefined slope represents a vertical line. Understanding slope is crucial for graphing lines and analyzing functions, as it provides insight into the behavior of the line, such as growth or decline, using expressions like \( y = 2(x + 10) \), where \( m = 2 \) suggests a constant rise over run.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, provides a connection between algebra and geometry using a coordinate system. It allows for precise and mathematical descriptions of geometrical shapes by utilizing points defined in the Cartesian plane. At the heart of coordinate geometry are concepts like coordinates, lines, and slopes which help solve geometric problems. Coordinate geometry enables various tasks, such as finding the distance between points, identifying midpoints, and formulating equations of geometrical figures like circles and lines.
In practical terms, coordinate geometry helps bridge numerics and shape understanding:

• Coordinates: Numerical expressions in the form of \((x, y)\) that pinpoint locations on the Cartesian plane.
• Lines and slopes: Understanding the characteristics of lines through equations and slopes aids in interpreting and predicting line behavior.
• Distance and midpoints: Calculating these offers insight into the spatial relationship between points.

This mathematical field is crucial for everything from basic line equations to complex geometrical constructions. By working with an example like formulating the line through \((-10, 0)\) with a slope of \(2\), coordinate geometry demonstrates its efficacy in simplifying and solving spatial problems.