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Chapter 4: Problem 434

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (-1,3) and (-6,-7)

Short Answer

Expert verified
The equation of the line is y = 2x + 5.

Step by step solution

01

Identify the coordinates

The given points are \((-1, 3)\) and \((-6, -7)\). Label these as \(x_1 = -1, y_1 = 3\) and \(x_2 = -6, y_2 = -7\).
02

Calculate the slope

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the values, we get: \[ m = \frac{-7 - 3}{-6 - (-1)} = \frac{-10}{-5} = 2 \] Therefore, the slope \(m = 2\).
03

Use point-slope form

The point-slope formula is \(y - y_1 = m(x - x_1)\). Choose one of the given points, let's use \((-1, 3)\). Substituting the values, we get: \[ y - 3 = 2(x + 1) \]
04

Simplify to slope-intercept form

Distribute and rearrange to get the equation in slope-intercept form. Starting with \[ y - 3 = 2(x + 1) \], distribute the 2: \[ y - 3 = 2x + 2 \] Add 3 to both sides: \[ y = 2x + 5 \] Therefore, the equation in slope-intercept form is \( y = 2x + 5 \).

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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 is one of the most common ways to express the equation of a line. It is written as:
y = mx + b , where
  • y is the dependent variable (usually on the vertical axis).
  • x is the independent variable (usually on the horizontal axis).
  • m is the slope of the line.
  • b is the y-intercept of the line, the point where the line crosses the y-axis.
To find the equation of a line, we first need the slope (m) and the y-intercept (b). Often, we derive the slope from two points on the line, and then use one of those points to find the y-intercept.
slope calculation
To determine the slope of a line, we use the formula:
m = \frac{y_2 - y_1}{x_2 - x_1} . Here,
  • (x1, y1) and (x2, y2) are two points on the line.
In our example, the coordinates are (-1, 3) and (-6, -7). Plugging these values into the slope formula gives us:
  • \[ m = \frac{-7 - 3}{-6 - (-1)} \]
  • \[ m = \frac{-10}{-5} \]
  • m = 2
The slope, m, tells us the line rises by 2 units for every 1 unit it moves to the right. A positive slope means the line ascends, while a negative slope means it descends.
point-slope form
Once we have the slope, we can use the point-slope form to write the equation of the line. The point-slope formula is:
y - y_1 = m(x - x_1) , where
  • (x1, y1) is a point on the line.
  • m is the slope.
In our example, using the point (-1, 3) and the slope (m = 2), we substitute these values into the formula to get:
  • y - 3 = 2(x + 1)
This form is useful because it allows for quick substitution of the slope and a specific point on the line, making it easier to derive the line's equation.
coordinate geometry
Coordinate geometry allows us to describe geometrical shapes and relationships using algebra. By placing points on a coordinate plane and connecting them with straight lines, we can form and analyze geometric shapes.
Some important aspects include:
  • Points: Defined by coordinates \(x, y\) on the coordinate plane.
  • Slope: Describes the steepness and direction of a line. Positive slopes go upwards, and negative slopes go downwards.
  • Distance: The length between two points, found using the distance formula: \[ d = \sqrt{{(x2 - x1)^2 + (y2 - y1)^2}} \]
  • Midpoint: The midpoint of a line segment connecting two points is found using: \[ M = \left( \frac{x1+x2}{2}, \frac{y1+y2}{2} \right) \]
These basic concepts help in understanding more complex geometrical properties and relationships using algebra.

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