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Magazine Chapter 3: Problem 61
Find the equation of a line, given the slope and a point on the line. $$ m=1 / 2 ;(-4,8) $$
Short Answer
Expert verified
The equation of the line is \( y = \frac{1}{2}x + 10 \).
Step by step solution
01
- Understand the given information
We are tasked with finding the equation of a line. We know the slope \( m \) is \( \frac{1}{2} \) and the line passes through the point \((-4, 8)\). We will use the point-slope formula to find the equation.
02
- Use the Point-Slope Formula
The point-slope formula is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \((x_1, y_1)\) is a point on the line. Substitute \( m = \frac{1}{2} \), \( x_1 = -4 \), and \( y_1 = 8 \) into the formula.
03
- Plug Values into the Formula
Substitute the values into the equation: \[ y - 8 = \frac{1}{2}(x + 4) \]This equation represents the line in point-slope form.
04
- Simplify to Slope-Intercept Form
Simplify the equation into the slope-intercept form (\( y = mx + b \)):1. Distribute the slope on the right: \( y - 8 = \frac{1}{2}x + \frac{1}{2}(4) \)2. Simplify: \( y - 8 = \frac{1}{2}x + 2 \)3. Add 8 to both sides: \( y = \frac{1}{2}x + 10 \).
05
- Verify the Equation
Verify that the point \((-4, 8)\) satisfies the equation \( y = \frac{1}{2}x + 10 \):Substitute \( x = -4 \), \( y = 8 \) into the equation:\[ 8 = \frac{1}{2}(-4) + 10 \]\[ 8 = -2 + 10 \]\[ 8 = 8 \]The point satisfies the equation, confirming it is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form of a linear equation is a convenient way to express the equation of a line. It is written as \( y = mx + b \). In this formula:
- \( m \) represents the slope of the line, which shows how steep the line is.
- \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
Linear Equation
A linear equation forms a straight line when plotted on a graph. It's characterized by its constant slope and simple structure. Linear equations have no variables with exponents, making them straightforward to analyze and graph.Key properties of linear equations include:
- An equation of the form \( y = mx + b \) is always a straight line.
- The slope \( m \) determines the angle of the line: a positive slope ascends from left to right, while a negative slope descends.
- The y-intercept \( b \) shows the initial position or the point where the line crosses the y-axis.
Coordinate Geometry
Coordinate geometry, or analytic geometry, bridges algebra with geometry using a coordinate system. It allows us to find the relationships between points, lines, and figures using numeric coordinates.In coordinate geometry, each point is defined by a pair of numbers \((x, y)\). Lines and curves can be plotted using coordinates to find intersections, parallelism, and other relationships.With coordinates:
- We can easily calculate the distance between two points using the distance formula.
- The midpoint formula helps find the center of a line segment between two points.
- Equations of lines like the slope-intercept and point-slope forms allow us to explore linear relationships effortlessly.
