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Chapter 2: Problem 27

Exer. 21-32: Find a general form of an equation of the line through the point \(A\) that satisfies the given condition. $$A(4,-5) ; \quad \text { through } B(-3,6)$$

Short Answer

Expert verified
The general form of the line is \(11x + 7y = 9\).

Step by step solution

01

Identify Given Points

The problem provides two points, \(A(4, -5)\) and \(B(-3, 6)\). These are the coordinates we will use to derive the line's equation.
02

Calculate the Slope

Use the formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\): \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the values: \(m = \frac{6 - (-5)}{-3 - 4} = \frac{11}{-7}\). So, the slope \(m = -\frac{11}{7}\).
03

Use Point-Slope Form

Now, use the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\). Substituting one point (e.g., point \(A(4, -5)\)) and the slope \(m = -\frac{11}{7}\), the equation becomes: \(y + 5 = -\frac{11}{7}(x - 4)\).
04

Simplify to General Form

Simplify the equation from Step 3 to reach the general form \(Ax + By = C\). Start by distributing: \(y + 5 = -\frac{11}{7}x + \frac{44}{7}\). Multiply through by 7 to clear the fraction: \(7y + 35 = -11x + 44\). Rearrange to standard form: \(11x + 7y = 9\).

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Key Concepts

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

Understanding Slope Calculation
The slope is a measure of how steep a line is. It tells us how much the line rises or falls between two points. To calculate the slope of a line through any two points, you can use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here, \(x_1\) and \(y_1\) are the coordinates of the first point, and \(x_2\) and \(y_2\) are the coordinates of the second point.
Let's break down the calculation using our example:
  • Identify points \(A(4, -5)\) and \(B(-3, 6)\).
  • Substitute these into the formula: \( m = \frac{6 - (-5)}{-3 - 4} \).
  • This simplifies to \( m = \frac{11}{-7} \).
  • Therefore, we find the slope \( m = -\frac{11}{7} \).
Understanding the slope is crucial because it forms the basis for writing the equation of the line in various forms.
Using the Point-Slope Form
The point-slope form is useful for writing the equation of a line when you know one point it passes through and its slope.
This form is written as follows:
  • \( y - y_1 = m(x - x_1) \)
Here's how you can apply this with a practical example:
Given a point \(A(4, -5)\) and slope \(m = -\frac{11}{7}\), plug these into the point-slope formula:
  • \( y + 5 = -\frac{11}{7}(x - 4) \)
This expresses the relationship between \(x\) and \(y\).
It's often a preferred starting point before converting into other forms, like the general form of a line.
Note how we substituted \(x_1 = 4\) and \(y_1 = -5\) into the formula, ensuring the equation accurately represents the line through the specified point with the given slope.
Deriving the General Form of a Line
The general form of a line is expressed as:
  • \(Ax + By = C\)
It's a more formal way of representing a linear equation, eliminating fractions and making computations straightforward.
Starting with the equation in point-slope form, as derived in the previous step, our task is to convert it into this general format through simplification:
  • From the point-slope form: \( y + 5 = -\frac{11}{7}(x - 4) \).
  • Distribute the slope: \( y + 5 = -\frac{11}{7}x + \frac{44}{7} \).
  • Eliminate the fraction by multiplying through by 7: \(7(y + 5) = -11x + 44 \).
  • This becomes \(7y + 35 = -11x + 44 \).
  • Rearrange to get the standard form: \( 11x + 7y = 9 \).
Achieving the general form involves making sure all terms are integers, and it presents the equation in a cleaner fashion.
This format is particularly useful in various mathematical applications, including systems of equations or when plotting in coordinate geometry.

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