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Chapter 3: Problem 35

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. $$ (2,3) \text { and }(-1,-1) $$

Short Answer

Expert verified
The equation of the line is \(4x - 3y = -1\).

Step by step solution

01

Identify the formula for the slope between two points

The formula for calculating the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In this case, the points are \((2, 3)\) and \((-1, -1)\). We will use this formula to find the slope.
02

Calculate the slope

Substitute the coordinates \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (-1, -1)\) into the slope formula: \[ m = \frac{-1 - 3}{-1 - 2} = \frac{-4}{-3} = \frac{4}{3} \] The slope of the line is \( \frac{4}{3} \).
03

Use point-slope formula to find equation of the line

We use the point-slope form of the equation, which is given by: \[ y - y_1 = m(x - x_1) \] Using the point \((2, 3)\) and the slope \( \frac{4}{3} \), we substitute into the equation: \[ y - 3 = \frac{4}{3}(x - 2) \]
04

Simplify the point-slope equation to standard form

First, simplify the point-slope equation: \[ y - 3 = \frac{4}{3}x - \frac{8}{3} \] Add 3 to both sides: \[ y = \frac{4}{3}x - \frac{8}{3} + \frac{9}{3} \] \[ y = \frac{4}{3}x + \frac{1}{3} \] Now we convert this to the standard form: Multiply through by 3 to eliminate fractions: \[ 3y = 4x + 1 \] Rearrange to get \(Ax + By = C\) form: \[ 4x - 3y = -1 \]
05

Write the final equation in standard form

The standard form of the line's equation is \( 4x - 3y = -1 \). This represents the line passing through points (2,3) and (-1,-1).

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Key Concepts

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

Slope of a Line
The slope of a line is a measure of its steepness, typically represented by the letter \( m \). It indicates how much the line rises or falls as you move from left to right. Finding the slope involves using the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
In this formula, \((x_1, y_1)\) and \((x_2, y_2)\) are points on the line. The numerator \( (y_2 - y_1) \) represents the change in \( y \), and the denominator \( (x_2 - x_1) \) represents the change in \( x \). This ratio shows the rise over run between two points on the line. When you calculate the slope between points \((2, 3)\) and \((-1, -1)\), you apply:
  • \( m = \frac{-1 - 3}{-1 - 2} = \frac{4}{3} \)
Resulting in a positive value, indicating an upward slope.
Point-Slope Form
The point-slope form is a convenient way to express the equation of a line using one of its points, alongside its slope. The equation is structured as:
  • \( y - y_1 = m(x - x_1) \)
Here, \( (x_1, y_1) \) is a specific point on the line, and \( m \) is the slope. This form is useful for constructing the equation when you have one point and the slope ready. For our exercise:
  • Given point: \((2, 3)\)
  • Slope: \( \frac{4}{3} \)
Plug into point-slope form:
  • \( y - 3 = \frac{4}{3}(x - 2) \)
Point-slope form allows for straightforward conversion to other forms, such as standard form.
Standard Form of a Line
The standard form of a line's equation is a powerful tool, especially in situations where you need a uniform representation of multiple lines. Its format is:
  • \( Ax + By = C \)
\( A \), \( B \), and \( C \) are integers, with \( A \) typically positive for standardization. To convert from point-slope to standard form, you rearrange and simplify the equation. From our solved example:
  • Started with: \( y = \frac{4}{3}x + \frac{1}{3} \)
  • Eliminate fractions: \( 3y = 4x + 1 \)
  • Rearrange to: \( 4x - 3y = -1 \)
Standard form provides a clear-cut format that is easy to read and interpret, especially suitable for comparing different lines or solving systems of linear equations.

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Most popular questions from this chapter

Solve each equation for \(y.\) \(y-(-3)=4(x-(-5))\)

Solve. Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. See Example 8 . In 2006 , there were 935 thousand eating establishments in the United States. In 1996 , there were 457 thousand eating estab. lishments. (Source: National Restaurant Association) a. Write an equation describing the relationship between time and number of eating establishments. Use ordered pairs of the form (years past 1996, number of eating establishments in thousands). b. Use this equation to predict the number of eating establishments in 2010

Solve each equation for \(y .\) See Section 2.5 \(5 x+2 y=7\)

Find the value of \(x^{2}-3 x+1\) for each given value of \(x .\) See Section 1.7. $$ -1 $$

Example $$\text { If } f(x)=x^{2}+2 x+1, \text { find } f(\pi)$$ Solution: $$\begin{aligned} &f(x)=x^{2}+2 x+1\\\ &f(\pi)=\pi^{2}+2 \pi+1 \end{aligned}$$ Given the following functions, find the indicated values. $$f(x)=x^{2}-12$$ a. \(f(12)\) b. \(f(a)\)
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