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Chapter 3: Problem 32

Find an equation of the line passing through each pair of points. Write the equation in the form \(A x+B y=C .\) See Example 5. $$ (6,2) \text { and }(8,8) $$

Short Answer

Expert verified
The equation is \(3x - y = 16\).

Step by step solution

01

Determine the Slope

To find the equation of the line passing through two points, first compute the slope, \(m\), using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1) = (6, 2)\) and \((x_2, y_2) = (8, 8)\). Substituting in these values gives: \[ m = \frac{8 - 2}{8 - 6} = \frac{6}{2} = 3 \]. Thus, the slope \(m\) is 3.
02

Use Point-Slope Form

Use the point-slope formula to write the equation of the line: \[ y - y_1 = m(x - x_1) \]. Using \(m = 3\) and \((x_1, y_1) = (6, 2)\), the equation becomes \[ y - 2 = 3(x - 6) \].
03

Simplify to Slope-Intercept Form

Distribute and simplify the equation to move into slope-intercept form \(y = mx + b\): \[ y - 2 = 3x - 18 \]. Adding 2 to both sides gives \[ y = 3x - 16 \].
04

Convert to Standard Form

The standard form of a line's equation is \(Ax + By = C\). Rearrange \(y = 3x - 16\) into this form by moving all terms to one side: \[-3x + y = -16 \]. To avoid negative coefficients where possible, multiply the entire equation by -1: \[ 3x - y = 16 \].

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Key Concepts

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

Slope
When learning about the equation of a line, the concept of slope is foundational. The slope, often denoted as \( m \), measures the steepness of a line. It tells us how much \( y \) changes for a given change in \( x \). The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula essentially gives the vertical change (rise) divided by the horizontal change (run).
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • If the slope is zero, the line is horizontal.
  • If the line is vertical, the slope is undefined.
In the given exercise, using the points \((6, 2)\) and \((8, 8)\), the slope is calculated as \( 3 \). This tells us that for every 1 unit increase in \( x \), \( y \) increases by 3 units.
Point-Slope Form
The point-slope form is incredibly useful for writing the equation of a line when you have a point and a slope. The form of the point-slope equation is: \[ y - y_1 = m(x - x_1) \] This equation uses:
  • \((x_1, y_1)\) - a known point on the line.
  • \( m \) - the slope of the line.
It's especially handy because it directly incorporates the slope and a specific point. Once you've calculated the slope, you can quickly see how changes in \( x \) will affect \( y \). For the current exercise, using \( m = 3 \) and point \((6,2)\), the equation becomes \( y - 2 = 3(x - 6) \). This form illustrates the relationship between \( x \) and \( y \) with the given point and slope.
Slope-Intercept Form
The slope-intercept form is perhaps the most popular way to express the equation of a line for its simplicity and clarity. The form is: \[ y = mx + b \] Here, \( m \) is the slope, and \( b \) is the y-intercept 鈥� the point where the line crosses the y-axis. This form is perfect for quickly graphing a line. Just start at the y-intercept (\( b \)) and use the slope \( m \) to find other points. By converting \( y - 2 = 3(x - 6) \) into slope-intercept form, we distribute and simplify to find:
  • Original: \( y - 2 = 3x - 18 \)
  • Add 2 to both sides: \( y = 3x - 16 \)
Now, you can clearly see that the slope of the line is 3, and the line crosses the y-axis at \( -16 \).
Standard Form
The standard form of a line's equation provides a neat, general way to express lines, especially helpful for linear equations in different mathematical contexts. The standard form is given by: \[ Ax + By = C \] Here, \( A \), \( B \), and \( C \) are integers, and the coefficients are typically adjusted to keep \( A \) non-negative. Converting from slope-intercept form \( y = 3x - 16 \) to standard form involves rearranging the terms:
  • Original: \( y = 3x - 16 \)
  • Rearrange: \(-3x + y = -16 \)
  • To avoid negative coefficients, multiply by \(-1\): \( 3x - y = 16 \)
This form is versatile in that it can be used to easily identify whole values of \( x \) and \( y \) that satisfy the equation, which is why it is often used in systems of equations.

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Most popular questions from this chapter

Solve. Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. See Example 8 . In \(2002,\) crude oil production by OPEC countries was about 28.7 million barrels per day. In \(2007,\) crude oil production had risen to about 34.5 million barrels per day. (Source: OPEC) a. Write two ordered pairs of the form (years after 2002 , crude oil production) for this situation. b. Assume that crude oil production is linear between the years 2002 and 2007 . Use the ordered pairs from part (a) to write an equation of the line relating year and crude oil production. c. Use the linear equation from part (b) to estimate the crude oil production by OPEC countries in 2004

Write an equation in standard form of the line that contains the point \((-1,2)\) and is a. parallel to the line \(y=3 x-1\) b. perpendicular to the line \(y=3 x-1\)

Find the slope of the line that is (a) parallel and (b) perpendicular to the line through each pair of points. \((6,-2)\) and \((1,4)\)

Write an ordered pair for each point described. Point \(C\) is four units to the right of the \(y\) -axis and seven units Thelow the \(x\) -axis.

Write an ordered pair for each point described. Point \(D\) is three units to the left of the origin.
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