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Chapter 1: Problem 48

Find an equation of the line that passes through the point \((1,-2)\) and is perpendicular to the line passing through the points \((-2,-1)\) and \((4,3)\).

Short Answer

Expert verified
The equation of the line that passes through the point \((1, -2)\) and is perpendicular to the line passing through the points \((-2, -1)\) and \((4, 3)\) is: \(y = -\frac{1}{2}x - \frac{3}{2}\)

Step by step solution

01

Find the slope of the line passing through \((-2,-1)\) and \((4,3)\)

We will use the slope formula: \(\frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((-2, -1)\) as \((x_1, y_1)\) and \((4, 3)\) as \((x_2, y_2)\), we have: Slope = \(\frac{3 - (-1)}{4 - (-2)}\) = \(\frac{4}{6}\) Simplify the fraction to get the slope: Slope = \(\frac{2}{3}\)
02

Find the slope of the perpendicular line

To find the slope of the line perpendicular to the given line, we find the negative reciprocal of the slope we found in Step 1. Perpendicular slope = \(-\frac{1}{2}\)
03

Find the equation of the line in slope-intercept form

To find the equation of the line using the slope we found in Step 2 and the given point \((1, -2)\), we will use the point-slope form of a line: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is the point the line passes through. Substitute the values into the equation: \(y - (-2) = -\frac{1}{2}(x - 1)\) Simplify the equation: \(y + 2 = -\frac{1}{2}x + \frac{1}{2}\) Now, isolate y to put the equation in slope-intercept form: \(y = -\frac{1}{2}x - \frac{3}{2}\) The equation of the line that passes through the point \((1, -2)\) and is perpendicular to the line passing through the points \((-2, -1)\) and \((4, 3)\) is: \(\boxed{y = -\frac{1}{2}x - \frac{3}{2}}\)

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Key Concepts

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

Slope Calculation
Finding the slope of a line is key to understanding its steepness and direction. When you're given two points on a line, like
  • the points i.e., dealing with an exercise originating points (-2,-1) and you can use them 4,3)
, to determine the slope. The slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) helps you calculate this.
First, identify your points as \((x_1, y_1)\) and \((x_2, y_2)\). In this example, \((x_1, y_1) = (-2, -1)\) and \((x_2, y_2) = (4, 3)\).
Next, plug these coordinates into the formula:
  • \[ m = \frac{3 - (-1)}{4 - (-2)} \]
Perform the calculations:
  • \[ m = \frac{4}{6} \]
Simplify the fraction:
  • \[ m = \frac{2}{3} \]
Your slope is \( \frac{2}{3} \). Remember, the slope tells us how much the line goes up (or down if negative) for each step to the right. Since this slope is positive, the line rises from left to right.
Point-Slope Form
When you know a line's slope and a point on the line, you can use the point-slope form to write its equation.
  • The formula looks like this: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the known point.
In this exercise, the point provided is \((1, -2)\), and the slope is the perpendicular slope we calculated earlier: \(-\frac{1}{2}\).
Plug these into the point-slope form:
  • \( y - (-2) = -\frac{1}{2}(x - 1) \)
This equation represents the line before any further simplification. It's a great way to start finding the line equation, especially when you have quick access to a point and slope.
Line Equation
To find a line's equation in slope-intercept form, begin with the point-slope form and rearrange it to solve for \( y \). The slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Starting with \( y + 2 = -\frac{1}{2}x + \frac{1}{2} \), subtract 2 from each side to isolate \( y \):
  • \( y = -\frac{1}{2}x + \frac{1}{2} - 2 \)
Simplify further:
  • \( y = -\frac{1}{2}x - \frac{3}{2} \)
Now you have the line's equation. This form clearly shows how the y-value changes with x and gives a neat representation of the line's behavior. Whether graphed or used in calculations, the slope-intercept form is straightforward and efficient.

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Most popular questions from this chapter

The number of U.S. broadband Internet households (in millions) between the beginning of \(2004(t=0)\) and the beginning of \(2008(t=4)\) was estimated to be $$ f(t)=6.5 t+33 \quad(0 \leq t \leq 4) $$ Over the same period, the number of U.S. dial-up Internet households (in millions) was estimated to be $$ g(t)=-3.9 t+42.5 \quad(0 \leq t \leq 4) $$ a. Sketch the graphs of \(f\) and \(g\) on the same set of axes. b. Solve the equation \(f(t)=g(t)\) and interpret your result.

For each supply equation, where \(x\) is the quantity supplied in units of 1000 and \(p\) is the unit price in dollars, (a) sketch the supply curve and (b) determine the number of units of the commodity the supplier will make available in the market at the given unit price. $$ p=2 x+10 ; p=14 $$

With computer security always a hot-button issue, demand is growing for technology that authenticates and authorizes computer users. The following table gives the authentication software sales (in billions of dollars) from 1999 through \(2004(x=0\) represents 1999): $$ \begin{array}{ccccccc} \hline \text { Year, } \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Sales, } \boldsymbol{y} & 2.4 & 2.9 & 3.7 & 4.5 & 5.2 & 6.1 \\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the result of part (a) to estimate the sales for 2007 , assuming the trend continues.

Find an equation of the line that passes through the point \((-2,2)\) and is parallel to the line \(2 x-4 y-8=0\).

The Social Security (FICA) wage base (in thousands of dollars) from 2003 to 2008 is given in the accompanying table \((x=1\) corresponds to 2003): $$ \begin{array}{lccc} \hline \text { Year } & 2003 & 2004 & 2005 \\ \hline \text { Wage Base, } \boldsymbol{y} & 87 & 87.9 & 90.0 \\ \hline \\ \hline \text { Year } & 2006 & 2007 & 2008 \\ \hline \text { Wage Base, } \boldsymbol{y} & 94.2 & 97.5 & 102.6 \\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the result of part (a) to estimate the FICA wage base in 2012
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