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Chapter 2: Problem 9

Find the slope of the line through the pair of points, then determine the equation. a. (2,3) and (5,3) c. (-3,8) and (-3,4) b. (-4,-7) and (12,-7) d. (2,-3) and (2,-1)

Short Answer

Expert verified
a. y = 3 b. y = -7 c. x = -3 d. x = 2

Step by step solution

01

Understanding the slope formula

The slope of a line through two points ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewline ... ys two points given (x_1, y_1) and (x_2, y_2) is determined using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
02

Calculating the slope for each pair of points

Probability is calculated using the formula: \[ m = \frac{3 - 3}{5 - 2} = \frac{0}{3} = 0 \] Thus, the slope for part (a) is m = 0.For part (c), m = \frac{4 - 8}{-3 + 3} = \frac{-4}{0}.The slope is undefined because the denominator is zero.For part (b), m = \frac{-7 - (-7)}{12 - (-4)} = \frac{0}{16} = 0.Thus, the slope for part (b) is m = 0.For part (d), m = \frac{-1 - (-3)}{2 - 2} = \frac{2}{0}The slope is undefined because the denominator is zero.
03

Determining the equation of the line

To find the equation of the line, use the slope-intercept form: y = mx + b. To find b, plug in one of the points and solve for b.Part (a): The slope is 0, so y = 0x + b or y = b. Using point (2, 3), then 3 = b, so y = 3.Part (c): The slope is undefined, so the line is vertical at x = -3.Part (b): The slope is 0, so y = 0x + b or y = b. Using point (-4, -7), then -7 = b, so y = -7.Part (d): The slope is undefined, so the line is vertical at x = 2.

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Key Concepts

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

linear equations
A linear equation represents a straight line on a graph. The standard form of a linear equation is given by \ y = mx + b \. Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis. Linear equations are fundamental in algebra and are useful in various applications like predicting trends, solving problems, and modeling real-life situations. Understanding the components of linear equations is crucial as it helps in graphing the line and interpreting its behavior accurately.
slope formula
The slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the slope formula: \ \( m = \frac{y_2 - y_1}{x_2 - x_1} \) \ This formula calculates the 'rise' over the 'run,' or the change in the y-values divided by the change in the x-values. The slope indicates the steepness and direction of the line. For example:
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
  • A slope of zero indicates a horizontal line.
  • An undefined slope indicates a vertical line.
This formula is essential as it helps in determining and understanding the orientation of the line between any two points on a graph.
undefined slope
An undefined slope occurs when you have a vertical line. This happens when the x-coordinates of two points are the same, resulting in a zero in the denominator of the slope formula. In such cases, the formula becomes: \ \( m = \frac{y_2 - y_1}{x_2 - x_1} \) \ For example, with points \( (-3, 8) \) and \( (-3, 4) \), the calculation would be: \ \( m = \frac{4 - 8}{-3 + 3} = \frac{-4}{0} \) \ Here, division by zero is impossible and thus, the slope is undefined. Vertical lines have equations of the form \ x = c \, where \( c \) is the common x-coordinate of all points on the line. Understanding undefined slopes is important for recognizing and dealing with vertical lines on graphs.

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Most popular questions from this chapter

Which equation has the steepest slope? a. \(y=2-7 x\) b. \(y=2 x+7\) c. \(y=-2+7 x\)

Construct an equation and sketch the graph of its line with the given slope, \(m,\) and vertical intercept, \(b .\) (Hint: Find two points on the line.) a. \(m=2, b=-3\) b. \(m=-\frac{3}{4}, b=1\) c. \(m=0, b=50\)

Identify the slopes and the vertical intercepts of the lines with the given equations. a. \(y=3+5 x\) b. \(f(t)=-t\) c. \(y=4\) d. \(Q=35 t-10\) e. \(f(E)=10,000+3000 E\)

Calculate the slope and write an equation for the linear function represented by each of the given tables. a. $$ \begin{array}{cc} \hline x & y \\ \hline 2 & 7.6 \\ 4 & 5.1 \\ \hline \end{array} $$ b. $$ \begin{array}{cc} \hline A & W \\ \hline 5 & 12 \\ 7 & 16 \\ \hline \end{array} $$

a. Write an equation for the line parallel to \(y=4-x\) that passes through the point (3,7) . b. Find an equation for the line perpendicular to \(y=4-x\) that passes through the point (3,7) .
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