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Chapter 3: Problem 74

Find the equation of each line. Write the equation in standard form unless indicated otherwise. Through \((-4,-2)\) and \((-6,5) ;\) use function notation.

Short Answer

Expert verified
The equation in function notation is \(f(x) = -\frac{7}{2}x - 16\).

Step by step solution

01

Find the Slope of the Line

To find the slope \( m \) of the line passing through the points \((-4, -2)\) and \((-6, 5)\), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points: \[ m = \frac{5 - (-2)}{-6 - (-4)} = \frac{5 + 2}{-6 + 4} = \frac{7}{-2} = -\frac{7}{2} \] So, the slope of the line is \( -\frac{7}{2} \).
02

Use Point-Slope Form

Now that we have the slope, we can use the point-slope form of the equation, \( y - y_1 = m(x - x_1) \). We will use the point \((-4, -2)\) and the slope \(-\frac{7}{2}\): \[ y - (-2) = -\frac{7}{2}(x - (-4)) \] Simplify it: \[ y + 2 = -\frac{7}{2}(x + 4) \]
03

Simplify to Function Notation

First, distribute \(-\frac{7}{2}\) in the equation from Step 2: \[ y + 2 = -\frac{7}{2}x - 14 \] Then, solve for \(y\) in terms of \(x\) to express it as a function \(f(x)\): \[ y = -\frac{7}{2}x - 14 - 2 \] \[ y = -\frac{7}{2}x - 16 \] This is equivalent to \( f(x) = -\frac{7}{2}x - 16 \).
04

Convert to Standard Form (Optional)

To express the equation in standard form \( Ax + By = C \), rearrange \( -\frac{7}{2}x - y = 16 \). Multiply each term by 2 to eliminate the fraction: \[ 7x + 2y = -32 \]

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Key Concepts

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

Slope Calculation
To find the slope of a line, you use a simple formula that calculates the steepness or incline between two points on a line. The formula is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
This equation reminds us of how to change in the y-values divided by the change in x-values. It鈥檚 called "rise over run."

In the given exercise, the points are
  • \((-4, -2)\)
  • \((-6, 5)\)
By plugging them into the formula, the calculation is \( \frac{5 - (-2)}{-6 - (-4)} \), which simplifies to \( \frac{7}{-2} \) or \(-\frac{7}{2}\).
This gives you the slope \( m \) = \( -\frac{7}{2} \), indicating our line falls from left to right.
Point-Slope Form
Point-slope form uses one of the line points and the slope you've calculated to find the equation of the line. The standard expression of the point-slope form is:
  • \( y - y_1 = m(x - x_1) \)
It鈥檚 a straightforward way to write the line's equation without having to solve for \( y \) right away.

In the exercise, using the point
  • \((-4, -2)\)
and slope
  • \(-\frac{7}{2}\)
you substitute into the form to get \( y + 2 = -\frac{7}{2}(x + 4) \). This step is critical because it allows you to move into other forms like the slope-intercept or standard form with ease.
Function Notation
Function notation is a way of expressing the equation of a line in terms of a function. Instead of writing \( y \), the equation is expressed as \( f(x) \), which represents the output of the function based on input \( x \).

To convert our line equation into function notation, you'll first need it in the form \( y = mx + b \).
From the working step, the equation becomes \( y = -\frac{7}{2}x - 16 \). Thus, in function notation, this is written as:
  • \( f(x) = -\frac{7}{2}x - 16 \)
Function notation is handy for graphing and identifying the slope and y-intercept easily.
Standard Form of a Line
The standard form of a line's equation is usually given by \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) is typically non-negative.

The transformation of a line equation to this form requires rearranging and sometimes multiplying to eliminate any fractions.
Starting with our function form \( -\frac{7}{2}x - y = 16 \), you need to eliminate the fraction by multiplying everything by 2, resulting in:
  • \( 7x + 2y = -32 \)
This conversion makes it easier to handle calculations involving systems of equations and is preferred in many geometry problems.

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