/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 68 You are given a line and a point... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 68

You are given a line and a point which is not on that line. Find the line perpendicular to the given line which passes through the given point. . \(y=\frac{4-x}{3}, P(1,-1)\)

Short Answer

Expert verified
The equation of the line perpendicular to \(y = \frac{4-x}{3}\) passing through \(P(1,-1)\) is \(y = 3x - 4\).

Step by step solution

01

Identify the Slope of the Given Line

The given line equation is \(y = \frac{4-x}{3}\). Rewriting it in the slope-intercept form \(y = mx + b\), we get \(y = -\frac{1}{3}x + \frac{4}{3}\). Thus, the slope \(m\) of the given line is \(-\frac{1}{3}\).
02

Find the Perpendicular Slope

The slope of a line perpendicular to another is the negative reciprocal of the original slope. The given line's slope is \(-\frac{1}{3}\), so the perpendicular slope is \(3\) (i.e., the negative reciprocal is \( \frac{1}{-\left(-\frac{1}{3}\right)} = 3\)).
03

Use Point-Slope Form to Write the Equation

We use the point-slope form \(y - y_1 = m(x - x_1)\), with the point \((1,-1)\) and the perpendicular slope \(3\). Substitute into the formula: \(y - (-1) = 3(x - 1)\).
04

Simplify the Equation

Simplify the equation from step 3: \(y + 1 = 3x - 3\). Solving for \(y\), subtract \(1\) from both sides to find the intercept form: \(y = 3x - 4\).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a line is an equation that helps us understand the linear relationship between two variables, usually the coordinates on a graph. The formula for this form is written as \( y = mx + b \), where:
  • \( y \) represents the dependent variable.
  • \( m \) is the slope of the line.
  • \( x \) is the independent variable.
  • \( b \) is the y-intercept, or the point where the line crosses the y-axis.
This form is especially useful because it clearly shows the slope and intercept of the line, making it easy to graph and understand. In our exercise, rewriting \( y = \frac{4-x}{3} \) as \( y = -\frac{1}{3}x + \frac{4}{3} \) allows us to quickly see the slope \( -\frac{1}{3} \) and the y-intercept \( \frac{4}{3} \). This step is crucial for further calculations when working with line equations.
Negative Reciprocal
When two lines are perpendicular, their slopes are related by the concept of a negative reciprocal. If the slope of one line is \( m \), the slope of a line perpendicular to it will be \(-\frac{1}{m}\).
  • This means you flip the fraction (reciprocal) and change the sign (negative).
  • For example, for a slope \( m = -\frac{1}{3} \), the perpendicular slope will be \( 3 \).
This relationship holds because the product of the slopes of two perpendicular lines is always \(-1\). Understanding negative reciprocals is key when finding perpendicular lines, as seen in our step-by-step solution, where we determined the perpendicular slope by taking the negative reciprocal of \(-\frac{1}{3}\), which results in a slope of \( 3 \).
Point-Slope Form
The point-slope form of a line equation can be a handy tool when you know a point on the line and its slope. It is represented as \( y - y_1 = m(x - x_1) \), where:
  • \( (x_1, y_1) \) are the coordinates of a known point on the line.
  • \( m \) is the slope of the line.
This form is particularly useful for quickly writing equations when solving problems involving specific points and slopes, like perpendicular lines. In our example, given a point \( (1, -1) \) and a perpendicular slope \( 3 \), the line's equation is easily constructed as \( y - (-1) = 3(x - 1) \). This allows us to directly connect the slope and a specific point to find the equation of the new line.
Equation of a Line
Understanding how to derive the equation of a line is essential in algebra and geometry. This can be done using either the slope-intercept form or the point-slope form, depending on the information you have.
  • The slope-intercept form \( y = mx + b \) requires knowing the y-intercept, whereas the point-slope form \( y - y_1 = m(x - x_1) \) needs at least one point on the line and its slope.
For the line we determined, our use of point-slope form resulted in an equation \( y - (-1) = 3(x - 1) \). After simplifying, the equation transforms into the slope-intercept form \( y = 3x - 4 \), which is more convenient for graphing. Each form offers insights into different aspects of the line and is chosen based on the problem's requirements.

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Most popular questions from this chapter

Prove that if \(|f(x)|=|g(x)|\) then either \(f(x)=g(x)\) or \(f(x)=-g(x) .\) Use that result to solve the equations. \(|1-2 x|=|x+1|\)

The chart below contains a portion of the fuel consumption information for a 2002 Toyota Echo that I (Jeff) used to own. The first row is the cumulative number of gallons of gasoline that I had used and the second row is the odometer reading when I refilled the gas tank. So, for example, the fourth entry is the point (28.25,1051) which says that I had used a total of 28.25 gallons of gasoline when the odometer read 1051 miles. $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Gasoline Used } & & & & & & & & & & & \\ \text { (Gallons) } & 0 & 9.26 & 19.03 & 28.25 & 36.45 & 44.64 & 53.57 & 62.62 & 71.93 & 81.69 & 90.43 \\ \hline \text { Odometer } & & & & & & & & & & & \\\\\text { (Miles) } & 41 & 356 & 731 & 1051 & 1347 & 1631 & 1966 & 2310 & 2670 & 3030 & 3371 \\\\\hline\end{array}$$ Find the least squares line for this data. Is it a good fit? What does the slope of the line represent? Do you and your classmates believe this model would have held for ten years had I not crashed the car on the Turnpike a few years ago? (I'm keeping a fuel log for my 2006 Scion xA for future College Algebra books so I hope not to crash it, too.)

Graph the quadratic function. Find the \(x\) - and \(y\) -intercepts of each graph, if any exist. If it is given in general form, convert it into standard form; if it is given in standard form, convert it into general form. Find the domain and range of the function and list the intervals on which the function is increasing or decreasing. Identify the vertex and the axis of symmetry and determine whether the vertex yields a relative and absolute maximum or minimum. \(f(x)=-3 x^{2}+4 x-7\)

Using the energy production data given below $$\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 \\\\\hline \begin{array}{l}\text { Production } \\\\\text { (in Quads) }\end{array} & 35.6 & 42.8 & 63.5 & 67.2 & 70.7 & 71.2 \\\\\hline\end{array}$$ (a) Plot the data using a graphing calculator and explain why it does not appear to be linear. (b) Discuss with your classmates why ignoring the first two data points may be justified from a historical perspective. (c) Find the least squares regression line for the last four data points and comment on the goodness of fit. Interpret the slope of the line of best fit. (d) Use the regression line to predict the annual US energy production in the year 2010 . (e) Use the regression line to predict when the annual US energy production will reach 100 Quads.

Graph the function. Find the zeros of each function and the \(x\) - and \(y\) -intercepts of each graph, if any exist. From the graph, determine the domain and range of each function, list the intervals on which the function is increasing, decreasing or constant, and find the relative and absolute extrema, if they exist. \(f(x)=|x+4|\)
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