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Chapter 2: Problem 66

Find the equation of the line that passes through the following points: \((2 a, b)\) and \((a, b+1)\)

Short Answer

Expert verified
The equation of the line is \( y = -\frac{1}{a}x + (b + 2) \).

Step by step solution

01

Identify the Slope Formula

The formula for the slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
02

Substitute the Given Points into the Slope Formula

The coordinates of the points are \((x_1, y_1) = (2a, b)\) and \((x_2, y_2) = (a, b+1)\).Substitute these values into the slope formula:\[ m = \frac{(b+1) - b}{a - 2a} = \frac{1}{-a} = -\frac{1}{a} \]
03

Use the Point-Slope Form of a Line

The point-slope form equation of a line is:\[ y - y_1 = m(x - x_1) \]You can use either of the given points. Choose the point \((2a, b)\):\[ y - b = -\frac{1}{a}(x - 2a) \]
04

Simplify the Equation

Distribute the slope in the equation:\[ y - b = -\frac{1}{a}x + 2 \]Rearrange it to the standard form by adding b to both sides:\[ y = -\frac{1}{a}x + 2 + b \]
05

Write the Final Equation in Slope-Intercept Form

Combine constants on the right side:\( y = -\frac{1}{a}x + (b + 2) \) which is the equation of the line in slope-intercept form \( y = mx + c \).

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Key Concepts

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

Slope Formula
The slope formula is essential for finding the steepness or incline of a line that connects two given points. This formula is written as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( m \) represents the slope, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.Understanding the slope gives insight into how much the line rises or falls as it moves from one point to another.
  • A positive slope indicates the line ascends when moving from left to right.
  • A negative slope suggests the line descends as it goes from left to right.
  • A zero slope means the line is horizontal, implying no rise.
  • An undefined slope is the indication of a vertical line.
In the original solution, the slope was calculated by substituting the points \((2a, b)\) and \((a, b+1)\) into the formula, leading to the slope \( m = -\frac{1}{a} \). This negative value shows the line slopes downwards, moving left to right.
Point-Slope Form
The point-slope form is a technique used to write the equation of a line when you know its slope and a point on the line. The general representation is given by:\[ y - y_1 = m(x - x_1) \]where \((x_1, y_1)\) is the point on the line, and \( m \) is the slope.This form is particularly helpful because it directly ties the equation to a specific point and the slope, making it easy to understand graphically how the line behaves.
  • You have the freedom to choose any point from the two given points.
  • This form helps you transition into other forms like slope-intercept form easily.
By plugging the slope \(-\frac{1}{a}\) and using the point \((2a, b)\), the solution was simplified to \[ y - b = -\frac{1}{a}(x - 2a) \]. This starts to build the framework for our line, anchored firmly to both its slope and a known point.
Slope-Intercept Form
The slope-intercept form is possibly the most popular and straightforward form of a linear equation:\[ y = mx + c \]where \( m \) stands for the slope and \( c \) represents the y-intercept of the line.This form clearly shows the slope of the line and where it intersects the y-axis, making it extremely beneficial when graphing or comparing lines.
  • Easy to plot on a coordinate plane鈥攋ust start at the y-intercept \( c \) and follow the slope \( m \).
  • Great for quickly identifying parallel or perpendicular lines by comparing slopes.
In the problem from the textbook, after manipulating the point-slope form equation, it was expressed in slope-intercept form as \( y = -\frac{1}{a}x + (b + 2) \). This shows the change in \( b \), resulting in a line that crosses the y-axis at \( b + 2 \), following the same negative slope as calculated earlier.

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Most popular questions from this chapter

Sketch a line with a \(y\) -intercept of \((0,5)\) and slope \(-\frac{5}{2}\).

Graph the function \(f\) on a domain of \([-10, \quad 10] : \quad f(x)=0.02 x-0.01 .\) Enter the function in a graphing utility. For the viewing window, set the minimum value of \(x\) to be \(-10\) and the maximum value of \(x\) to be \(10 .\)

Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in Table 2.17. Plot the points, then sketch a line that fits the data. $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Predicted } & {6} & {7} & {7} & {8} & {7} & {9} & {10} & {10} \\ \hline \text { Actual } & {6} & {7} & {8} & {8} & {9} & {10} & {10} & {9} \\ \hline\end{array}$$

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years: $$(2500,2000),(2650,2001),(3000,2003),(3500,2006),(4200,2010)$$ Predict when the population will hit 8,000.

Does the following table represent a linear function? If so, find the linear equation that models the data. $$\begin{array}{|c|c|c|c|c|}\hline x & {-4} & {0} & {2} & {10} \\ \hline g(x) & {18} & {-2} & {-12} & {-52} \\ \hline\end{array}$$
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