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Chapter 6: Problem 8

Find the equation of the line passing through \(\mathrm{A}(2,-1)\) and \(\mathrm{B}(5,8)\)

Short Answer

Expert verified
The equation of the line is \( y = 3x - 7 \).

Step by step solution

01

Find the slope (m)

The slope of a line passing through two points can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Use points A(2,-1) and B(5,8). The coordinates are \(x_1 = 2, y_1 = -1, x_2 = 5, y_2 = 8\). Substitute these into the formula: \[ m = \frac{8 - (-1)}{5 - 2} = \frac{8 + 1}{3} = \frac{9}{3} = 3 \]. Thus, the slope \( m = 3 \).
02

Use point-slope form

The point-slope formula for a line is \( y - y_1 = m(x - x_1) \). We will use point A(2, -1) and the found slope \( m = 3 \) here. Substitute the values into the formula: \[ y - (-1) = 3(x - 2) \]. This simplifies to \[ y + 1 = 3(x - 2) \].
03

Simplify to slope-intercept form

Starting from the equation \( y + 1 = 3(x - 2) \), first distribute the slope on the right side: \( y + 1 = 3x - 6 \). Next, solve for \( y \) by subtracting 1 from both sides: \[ y = 3x - 6 - 1 \] which simplifies to \[ y = 3x - 7 \]. This is the slope-intercept form of the equation of the line.

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Key Concepts

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

Understanding the Slope
The slope of a line is a number that indicates the direction and steepness of the line. When you look at a straight line on a graph, the slope tells you how much the line goes up or down for every unit it goes across to the right. It is often represented by the letter \( m \).

To find the slope of a line passing through two points, you use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
This equation might look intimidating at first, but here's what each part means:
  • \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.
  • Subtract \( y_1 \) from \( y_2 \) to find out how much higher or lower Point B is compared to Point A.
  • Subtract \( x_1 \) from \( x_2 \) to see the horizontal distance between the two points.
For the original exercise, the slope \( m \) was calculated as 3, showing the line rises 3 units vertically for each unit it moves horizontally.
Exploring the Point-Slope Form
Once you have the slope of a line, you can use the point-slope form to write the equation of the line. This form is handy because it incorporates both the slope and a specific point on the line. The point-slope form of a line's equation is:
  • \( y - y_1 = m(x - x_1) \)
Where:
  • \( m \) is the slope we calculated previously.
  • \( (x_1, y_1) \) is a point on the line (often where you start plotting the line).
In our exercise, using Point A(2, -1) and the slope previously found, the equation became \( y + 1 = 3(x - 2) \). It is the foundational form before transitioning to a more visually understandable version, the slope-intercept form.
Transitioning to Slope-Intercept Form
The slope-intercept form is possibly the most popular way to express the equation of a line. It's in the format:
  • \( y = mx + b \)
Where:
  • \( m \) is the slope.
  • \( b \) is the y-intercept (where the line crosses the y-axis).
From the point-slope equation \( y + 1 = 3(x - 2) \), the goal is to solve for \( y \) to reveal \( y = 3x - 7 \) as the line's equation in slope-intercept form.

Here, \( m = 3 \), which matches the slope found earlier, and \( b = -7 \), indicating the line crosses the y-axis at -7. This makes it easy to graph since you can start at \( b \) on the y-axis and follow the slope to plot the line.

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A function is periodic with period 1 and is odd. Sketch a possible form of this function.

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