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Chapter 6: Problem 14

\text { Find the equation of the line that passes through }(2.2,4.7) \text { and }(6.8,-3.9) \text {. }

Short Answer

Expert verified
The equation of the line is \( y = -\frac{43}{23}x + 8.81 \).

Step by step solution

01

Calculate the Slope

The first step to find the equation of the line is to calculate the slope using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the given points \((x_1, y_1) = (2.2, 4.7)\) and \((x_2, y_2) = (6.8, -3.9)\), substitute the values to find \( m = \frac{-3.9 - 4.7}{6.8 - 2.2} = \frac{-8.6}{4.6} \). Simplify this to get \( m = -\frac{43}{23} \).
02

Use Point-Slope Form

Now that we have the slope, use the point-slope form equation \( y - y_1 = m(x - x_1) \) with one of the points, e.g., \((2.2, 4.7)\). Substitute the slope \( m = -\frac{43}{23} \) and the point into the equation: \( y - 4.7 = -\frac{43}{23}(x - 2.2) \).
03

Simplify to Slope-Intercept Form

To express the equation in slope-intercept form (\(y = mx + b\)), expand the equation: \( y - 4.7 = -\frac{43}{23}x + \frac{43 \times 2.2}{23} \). Simplify \( \frac{43 \times 2.2}{23} \) to find \( y = -\frac{43}{23}x + 4.11 + 4.7 \). Combine like terms to obtain \( y = -\frac{43}{23}x + 8.81 \).

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Key Concepts

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

Slope Calculation
The first step in finding the equation of a line is to calculate the slope, which is crucial for understanding how steep a line is on a graph. The slope tells us how much the line rises or falls as it moves from left to right. The formula for finding 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 our exercise, the points are \((2.2, 4.7)\) and \((6.8, -3.9)\). So, we substitute these into our formula:
  • \( m = \frac{-3.9 - 4.7}{6.8 - 2.2} \)
  • This simplifies to \( m = \frac{-8.6}{4.6} \)
  • Further simplify to get \( m = -\frac{43}{23} \)
This slope value of \(-\frac{43}{23}\) indicates the line descends as it moves from left to right.
Point-Slope Form
Once the slope is known, the next step is to use the point-slope form of a line equation. This form is extremely useful when you already know a point on the line and the slope. The point-slope form equation is:
  • \( y - y_1 = m(x - x_1) \)
In this formula, \((x_1, y_1)\) represents a point on the line, and \(m\) is the slope. Using the point \((2.2, 4.7)\) and the previously calculated slope \(-\frac{43}{23}\), we substitute these values into the equation:
  • \( y - 4.7 = -\frac{43}{23}(x - 2.2) \)
This equation describes the line passing through the point \((2.2, 4.7)\) with a slope of \(-\frac{43}{23}\). It's a flexible form that sets us up for deriving other forms of the line equation.
Slope-Intercept Form
The slope-intercept form is perhaps the most widely recognized way to express the equation of a line. It is written as \( y = mx + b \), where \(m\) represents the slope and \(b\) is the y-intercept (the point where the line crosses the y-axis). To convert the point-slope form to this form, we start by expanding and simplifying the equation:
  • Start with: \( y - 4.7 = -\frac{43}{23}(x - 2.2) \)
  • Expand to: \( y - 4.7 = -\frac{43}{23}x + \frac{43 \times 2.2}{23} \)
  • Perform the multiplication: \( \frac{43 \times 2.2}{23} = 4.11 \)
  • Rearrange to get: \( y = -\frac{43}{23}x + 4.7 + 4.11 \)
  • This simplifies to the final form: \( y = -\frac{43}{23}x + 8.81 \)
In this form, the slope-intercept equation highlights the slope \(-\frac{43}{23}\) and the y-intercept \(8.81\), giving clear insights into the graph's characteristics.

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