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Chapter 9: Problem 11

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. Through \((7,-1)\) and \((5,3)\)

Short Answer

Expert verified
The general form is \[ 2x + y - 13 = 0 \].

Step by step solution

01

- Find the Slope

To find the slope of the line passing through the points \( (7, -1) \) and \( (5, 3) \), use the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Substituting the coordinates \((x_1, y_1) = (7, -1) \) and \((x_2, y_2) = (5, 3) \), we get \[ m = \frac{3 - (-1)}{5 - 7} = \frac{4}{-2} = -2 \].
02

- Use Point-Slope Form

With the slope \( m = -2 \) and a point \( (7, -1) \), use the point-slope form of the equation \[ y - y_1 = m(x - x_1) \]. Substituting \((x_1, y_1) = (7, -1) \), we get \[ y - (-1) = -2(x - 7) \]. This simplifies to \[ y + 1 = -2(x - 7) \].
03

- Simplify to General Form

Simplify the equation to get the general form \[ Ax + By + C = 0 \]. Starting from \[ y + 1 = -2(x - 7) \], distribute on the right-hand side: \[ y + 1 = -2x + 14 \]. Now, move all terms to one side to get: \[ 2x + y - 13 = 0 \]. This is the general form of the equation.
04

- Graph the Line

To graph the line, plot the points \( (7, -1) \) and \( (5, 3) \) and draw a line passing through them. Ensure the line continues beyond these points for accuracy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Formula
The slope formula is pivotal in understanding how steep a line is and which direction it goes. It calculates the 'rise' over the 'run' between two points on a line. The slope, commonly denoted as \( m \), is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In simpler terms, the slope is the change in the y-values (vertical) divided by the change in the x-values (horizontal). For the points \( (7, -1) \) and \( (5, 3) \), substituting these into the formula gives us: \[ m = \frac{3 - (-1)}{5 - 7} = \frac{4}{-2} = -2 \]This negative slope indicates the line decreases as one moves from left to right.
Point-Slope Form
Once the slope is determined, we can use the point-slope form of a line equation to create an equation. This form is particularly useful as it easily incorporates a known slope and a point through which the line passes. The point-slope form is expressed as: \[ y - y_1 = m(x - x_1) \]With the slope \( m = -2 \) and the point \( (7, -1) \), the equation becomes: \[ y - (-1) = -2(x - 7) \]This simplifies to: \[ y + 1 = -2(x - 7) \]This form visually shows how the line behaves at that specific point and slope.
Graphing Lines
Graphing a line helps visualize the relationship between the x and y values. Starting with our derived equation \( y + 1 = -2(x - 7) \), it can be useful to convert to general form. Simplifying further: \[ y + 1 = -2(x - 7) \]expands to: \[ y + 1 = -2x + 14 \]Moving all terms to one side results in: \[ 2x + y - 13 = 0 \]To graph this, plot the points \( (7, -1) \) and \( (5, 3) \) on the coordinate plane and draw a line through them. This line extends infinitely in both directions, representing all solutions to the equation.
Geometry
Understanding the geometric concept of a line is essential in grasping how equations translate to visual representation. In geometry, a line is defined by its slope and a point or by two points. The slope shows the 'steepness', while specific points provide exact locations.
  • A positive slope rises from left to right.
  • A negative slope falls from left to right.
For our given points \( (7, -1) \) and \( (5, 3) \) with the slope \( -2 \), the geometric representation confirms the line descends as x increases. Making these connections between algebraic equations and geometric lines solidifies a foundational understanding of linear relationships in geometry.

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Most popular questions from this chapter

Find the missing coordinate so that each ordered pair is a solution to the equation. $$x+y+2=0$$ $$\text { (a) }(0, ?) ; \text { (b) }(?, 0) ; \text { (c) }(1, ?) ; \quad \text { (d) }(?,-2)$$

Find the midpoint of the line segment between the points given. $$(5,-3) \text { and }(-2,-1)$$

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