Distance Show that the distance between the point and the line is

In Exercises 1–4, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to MathGraphs.com.

In Exercises 1–4, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to MathGraphs.com.

In Exercises 1–4, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to MathGraphs.com.

In Exercises 1–4, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to MathGraphs.com.

Finding the Slope of a Line In Exercises 510, plot the pair of points and find the slope of the line passing through them.

In Exercises 5-10, plot the pair of points and find the slope of the line passing through them. (1,1), (-2,7)

In Exercises 5-10, plot the pair of points and find the slope of the line passing through them. (4,6), (4,1)

In Exercises 5-10, plot the pair of points and find the slope of the line passing through them. (3,-5), (5,-5)

In Exercises 5-10, plot the pair of points and find the slope of the line passing through them. \(\left(-\frac{1}{2}, \frac{2}{3}\right),\left(-\frac{3}{4}, \frac{1}{6}\right)\) Text Transcription: (-1 / 2, 2 / 3), (-3 / 4, 1 / 6)

In Exercises 5-10, plot the pair of points and find the slope of the line passing through them. \(\left(\frac{7}{8}, \frac{3}{4}\right),\left(\frac{5}{4},-\frac{1}{4}\right)\) Text Transcription: (7 / 8, 3 / 4), (5 / 4, -1 / 4)

In Exercises 11 and 12, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point (3,4) Slopes (a) 1 (b) -2 (c) \(-\frac{3}{2}\) (d) Undefined Text Transcription: - 3 / 2

In Exercises 11 and 12, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point (-2,5) Slopes (a) 3 (b) -3 (c) \(\frac{1}{3}\) (d) 0 Text Transcription: 1 / 3

In Exercises 13-16, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point Slope (6,2) \(m\) = 0 Text Transcription: m

In Exercises 13-16, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point Slope (-4,3) \(m\) is undefined. Text Transcription: m

In Exercises 13-16, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point Slope (1,7) \(m\) = - 3 Text Transcription: m

In Exercises 13-16, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point Slope (-2,-2) \(m\) = 2 Text Transcription: m

In Exercises 17-22, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope (0,3) \(m=\frac{3}{4}\) Text Transcription: m = 3 / 4

In Exercises 17-22, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope (-5,-2) \(m\) Text Transcription: m

In Exercises 17-22, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope (0,0) \(m=\frac{2}{3}\) Text Transcription: m = 2 / 3

In Exercises 17-22, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope (0,4) \(m\) = 0 Text Transcription: m

In Exercises 17-22, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope (3,-2) \(m\) = 3 Text Transcription: m

In Exercises 17-22, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope (-2,4) \(m=-\frac{3}{5}\) Text Transcription: m = - 3 / 5

A moving conveyor is built to rise 1 meter for each 3 meters of horizontal change. (a) Find the slope of the conveyor. (b) Suppose the conveyor runs between two floors in a factory. Find the length of the conveyor when the vertical distance between floors is 10 feet.

The table shows the populations \(y\) (in millions) of the United States for 2004 through 2009. The variable \(t\) represents the time in years, with corresponding to 2004. (Source: U.S. Census Bureau) (a) Plot the data by hand and connect adjacent points with a line segment. (b) Use the slope of each line segment to determine the year when the population increased least rapidly. (c) Find the average rate of change of the population of the United States from 2004 through 2009. (d) Use the average rate of change of the population to predict the population of the United States in 2020. Text Transcription: y t

In Exercises 25-30, find the slope and the \(y\)-intercept (if possible) of the line. y = 4x - 3 Text Transcription: y

In Exercises 25-30, find the slope and the \(y\)-intercept (if possible) of the line. -x + y = 1 Text Transcription: y

In Exercises 25-30, find the slope and the \(y\)-intercept (if possible) of the line. x + 5y = 20 Text Transcription: y

In Exercises 25-30, find the slope and the \(y\)-intercept (if possible) of the line. 6x - 5y = 15 Text Transcription: y

In Exercises 25-30, find the slope and the \(y\)-intercept (if possible) of the line. x = 4 Text Transcription: y

In Exercises 25-30, find the slope and the \(y\)-intercept (if possible) of the line. y = - 1 Text Transcription: y

In Exercises 31-38, sketch a graph of the equation. y = - 3

In Exercises 31-38, sketch a graph of the equation. x = 4

In Exercises 31-38, sketch a graph of the equation. y = - 2x + 1

In Exercises 31-38, sketch a graph of the equation. \(y=\frac{1}{3} x-1\) Text Transcription: y = 1 / 3x -1

In Exercises 31-38, sketch a graph of the equation. \(y-2=\frac{3}{2}(x-1)\) Text Transcription: y - 2 = 3 / 2 (x - 1)

In Exercises 31-38, sketch a graph of the equation. y - 1 = 3(x + 4)

In Exercises 31-38, sketch a graph of the equation. 2x - y - 3 = 0

In Exercises 31-38, sketch a graph of the equation. x + 2y + 6 = 0

In Exercises 39-46, find an equation of the line that passes through the points. Then sketch the line. (0,0), (4,8)

In Exercises 39-46, find an equation of the line that passes through the points. Then sketch the line. (-2,-2), (1,7)

In Exercises 39-46, find an equation of the line that passes through the points. Then sketch the line. (2, 8), (5, 0)

In Exercises 39-46, find an equation of the line that passes through the points. Then sketch the line. (-3,6), (1,2)

In Exercises 39-46, find an equation of the line that passes through the points. Then sketch the line. (6,3), (6,8)

In Exercises 39-46, find an equation of the line that passes through the points. Then sketch the line. (1,-2), (3,-2)

In Exercises 39-46, find an equation of the line that passes through the points. Then sketch the line. \(\left(\frac{1}{2}, \frac{7}{2}\right),\left(0, \frac{3}{4}\right)\) Text Transcription: (1 / 2, 7 / 2), (0, 3 / 4)

In Exercises 39-46, find an equation of the line that passes through the points. Then sketch the line. \(\left(\frac{7}{8}, \frac{3}{4}\right),\left(\frac{5}{4},-\frac{1}{4}\right)\) Text Transcription: (7 / 8, 3 / 4), (5 / 4, -1 / 4)

Find an equation of the vertical line with \(x\)-intercept at 3. Text Transcription: x

Show that the line with intercepts \((a, 0)\) and \((0, b)\) has the following equation. \(\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0\) Text Transcription: (a, 0) (0, b) x / a + y / b = 1, a neq 0, b neq 0

In Exercises 49-54, use the result of Exercise 48 to write an equation of the line in a general form. \(x\) -intercept: (2,0) \(y\)-intercept: (0,3) Text Transcription: x y

In Exercises 49-54, use the result of Exercise 48 to write an equation of the line in a general form. \(x\)-intercept: \(\left(-\frac{2}{3}, 0\right)\) \(y\)-intercept: (0,-2) Text Transcription: x -2 / 3, 0 y

In Exercises 49-54, use the result of Exercise 48 to write an equation of the line in a general form. Point on line: (1,2) \(x\)-intercept: \((a, 0)\) \(y\)-intercept: \((0, a)\) \((a \neq 0)\) Text Transcription: x (a, 0) y (0, a) (a neq 0)

In Exercises 49-54, use the result of Exercise 48 to write an equation of the line in a general form. Point on line: (-3,4) \(x\)-intercept: \((a, 0)\) \(y\)-intercept: \((0, a)\) \((a \neq 0)\) Text Transcription: x (a, 0) y (0, a) (a neq 0)

In Exercises 49-54, use the result of Exercise 48 to write an equation of the line in a general form. Point on line: (9,-2) \(x\)-intercept: \(2 a, 0)\) \(y\)-intercept: \((0, a)\) \((a \neq 0)\) Text Transcription: x (2 a, 0) y (0, a) (a neq 0)

In Exercises 49-54, use the result of Exercise 48 to write an equation of the line in a general form. Point on line: \(\left(-\frac{2}{3},-2\right)\) \(x\)-intercept: \((a, 0)\) \(y\)-intercept: \((0,-a)\) \((a \neq 0)\) Text Transcription: -2 / 3, -2 x (a, 0) y (0,-a) (a neq 0)

In Exercises 55-62, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line (-7,-2) x = 1

In Exercises 55-62, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line (-1,0) y = - 3

In Exercises 55-62, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line (2,5) x - y = - 2 Text Transcription: x - y

In Exercises 55-62, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line (-3,2) x + y = 7 Text Transcription: x + y

In Exercises 55-62, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line (2,1) 4x - 2y = 3

In Exercises 55-62, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line \(\left(\frac{5}{6},-\frac{1}{2}\right)\) 7x + 4 y = 8 Text Transcription: (5 / 6, -1 / 2)

In Exercises 55-62, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line \(\left(\frac{3}{4}, \frac{7}{8}\right)\) 5x - 3y = 0 Text Transcription: (3 / 4, 7 / 8)

In Exercises 55-62, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line (4,-5) 3x + 4y = 7

In Exercises 63-66, you are given the dollar value of a product in 2012 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value \(V\) of the product in terms of the year \(t\). (Let t = 0 represent 2010.) 2012 Value Rate $1850 $250 increase per year Text Transcription: V t

In Exercises 63-66, you are given the dollar value of a product in 2012 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value \(V\) of the product in terms of the year \(t\). (Let t = 0 represent 2010.) 2012 Value Rate $156 $4.50 increase per year Text Transcription: V t

In Exercises 63-66, you are given the dollar value of a product in 2012 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value \(V\) of the product in terms of the year \(t\). (Let t = 0 represent 2010.) 2012 Value Rate $17,200 $1600 decrease per year Text Transcription: V t

In Exercises 63-66, you are given the dollar value of a product in 2012 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value \(V\) of the product in terms of the year \(t\). (Let t = 0 represent 2010.) 2012 Value Rate $245,000 $5600 decrease per year Text Transcription: V t

In Exercises 67 and 68, determine whether the points are collinear. (Three points are collinear if they lie on the same line.) (-2,1), (-1,0), (2,-2)

In Exercises 67 and 68, determine whether the points are collinear. (Three points are collinear if they lie on the same line.) (0,4), (7,-6), (-5,11)

In Exercises 69–71, find the coordinates of the point of intersection of the given segments. Explain your reasoning.

In Exercises 69–71, find the coordinates of the point of intersection of the given segments. Explain your reasoning.

In Exercises 69–71, find the coordinates of the point of intersection of the given segments. Explain your reasoning.

Show that the points of intersection in Exercises 69, 70, and 71 are collinear.

Analyzing a Line A line is represented by the equation \(a x+b y=4\). (a) When is the line parallel to the \(x\)-axis? (b) When is the line parallel to the \(y\)-axis? (c) Give values for \(a\) and \(b\) such that the line has a slope of \(\frac{5}{8}\). (d) Give values for \(a\) and \(b\) such that the line is perpendicular to \(y=\frac{2}{5} x+3\). (e) Give values for \(a\) and \(b\) such that the line coincides with the graph of 5x + 6y = 8. Text Transcription: ax + by = 4 x y a b 5 / 8 y = 2 / 5x + 3

Several lines (labeled \(a – f\) ) are shown in the figure below. a) Which lines have a positive slope? (b) Which lines have a negative slope? (c) Which lines appear parallel? (d) Which lines appear perpendicular? Text Transcription: a - f

Find a linear equation that expresses the relationship between the temperature in degrees Celsius \(C\) and degrees Fahrenheit \(F\). Use the fact that water freezes at \(0^{\circ} \mathrm{C}\left(32^{\circ} \mathrm{F}\right)\) and boils at \(100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right)\). Use the equation to convert \(72^{\circ} \mathrm{F}\) to degrees Celsius. Text Transcription: C F (0^{circ}C (32^{circ}F) 100^{circ}C (212^{circ}F) 72^{circ}F

A company reimburses its sales representatives $200 per day for lodging and meals plus $0.51 per mile driven. Write a linear equation giving the daily cost \(C\) to the company in terms of \(x\), the number of miles driven. How much does it cost the company if a sales representative drives 137 miles on a given day? Text Transcription: C x

As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% of sales. You are offered a new job at $2300 per month, plus a commission of 5% of sales. (a) Write linear equations for your monthly wage in terms of your monthly sales \(s\) for your current job and your job offer. (b) Use a graphing utility to graph each equation and find the point of intersection. What does it signify? (c) You think you can sell $20,000 worth of a product per month. Should you change jobs? Explain. Text Transcription: s

A small business purchases a piece of equipment for $875. After 5 years, the equipment will be outdated, having no value. (a) Write a linear equation giving the value \(y\) of the equipment in terms of the time \(x\) (in years), \(0 \leq x \leq 5\). (b) Find the value of the equipment when x = 2. (c) Estimate (to two-decimal-place accuracy) the time when the value of the equipment is $200. Text Transcription: y x 0 leq x leq 5

A real estate office manages an apartment complex with 50 units. When the rent is $780 per month, all 50 units are occupied. However, when the rent is $825, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent \(p\) and the demand \(x\) is linear. (Note: The term demand refers to the number of occupied units.) (a) Write a linear equation giving the demand \(x\) in terms of the rent \(p\). (b) Linear extrapolation Use a graphing utility to graph the demand equation and use the trace feature to predict the number of units occupied when the rent is raised to $855. (c) Linear interpolation Predict the number of units occupied when the rent is lowered to $795. Verify graphically. Text Transcription: p x

An instructor gives regular 20-point quizzes and 100-point exams in a mathematics course. Average scores for six students, given as ordered pairs \((x, y)\), where \(x\) is the average quiz score and is the average exam score, are (18, 87), (10, 55), (19, 96), (16, 79), (13, 76) and (15, 82). (a) Use the regression capabilities of a graphing utility to find the least-squares regression line for the data. (b) Use a graphing utility to plot the points and graph the regression line in the same viewing window. (c) Use the regression line to predict the average exam score for a student with an average quiz score of 17. (d) Interpret the meaning of the slope of the regression line. (e) The instructor adds 4 points to the average exam score of everyone in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line. Text Transcription: (x, y) x

Find an equation of the line tangent to the circle \(x^{2}+y^{2}=169\) at the point (5,12). Text Transcription: x^{2} + y^{2} = 169

Find an equation of the line tangent to the circle \((x-1)^{2}+(y-1)^{2}=25\) at the point (4,-3). Text Transcription: (x - 1)^{2} + (y - 1)^{2} = 25

In Exercises 83-86, find the distance between the point and line, or between the lines, using the formula for the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+ C=0\) Distance \(=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Point: (- 2, 1) Line: x - y - 2 = 0 Text Transcription: (x_{1}, y_{1}) Ax + By + C = 0 = |Ax_{1} + By_{1} + C| / sqrt{A^{2}+B^{2}}

In Exercises 83-86, find the distance between the point and line, or between the lines, using the formula for the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+ C=0\) Distance \(=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Point: (2, 3) Line: 4x + 3y = 10 Text Transcription: (x_{1}, y_{1}) Ax + By + C = 0 = |Ax_{1} + By_{1} + C| / sqrt{A^{2}+B^{2}}

In Exercises 83-86, find the distance between the point and line, or between the lines, using the formula for the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+ C=0\) Distance \(=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Line: x + y = 1 Line: x + y = 5 Text Transcription: (x_{1}, y_{1}) Ax + By + C = 0 = |Ax_{1} + By_{1} + C| / sqrt{A^{2}+B^{2}}

In Exercises 83-86, find the distance between the point and line, or between the lines, using the formula for the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+ C=0\) Distance \(=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Line: 3x - 4y = 1 Line: 3x - 4y = 10 Text Transcription: (x_{1}, y_{1}) Ax + By + C = 0 = |Ax_{1} + By_{1} + C| / sqrt{A^{2}+B^{2}}

Show that the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+C=0\) is Distance \(=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\). Text Transcription: (x_{1}, y_{1}) Ax + By + C = 0 = |Ax_1 + By_1 + C| / sqrt{A^{2} + B^{2}

Write the distance \(d\) between the point (3,1) and the line y = mx + 4 in terms of \(m\). Use a graphing utility to graph the equation. When is the distance 0? Explain the result geometrically. Text Transcription: d m

Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.)

Prove that the figure formed by connecting consecutive midpoints of the sides of any quadrilateral is a parallelogram.

Prove that if the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) lie on the same line as \(\left(x_{1}^{*}, y_{1}^{*}\right)\) and \(\left(x_{2}^{*}, y_{2}^{*}\right)\), then \(\frac{y_{2}^{*}-y_{1}^{*}}{x_{2}^{*}-x_{1}^{*}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) . Assume \(x_{1} \neq x_{2}\) and \(x_{1}^{*} \neq x_{2}^{*}\). Text Transcription: (x_{1}, y_{1}) (x_{2}, y_{2}) (x_{1}^{*}, y_{1}^{*}) (x_{2}^{*}, y_{2}^{*}) {y_{2}^{*} - y_{1}^{*} / {x_{2}^{*} - x_{1}^{*} = {y_{2} - y_{1} / {x_{2} - x_{1} x_{1} neq x_{2} x_{1}^{*} neq x_{2}^{*}

Prove that if the slopes of two nonvertical lines are negative reciprocals of each other, then the lines are perpendicular.

In Exercises 93-96, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The lines represented by \(a x+b y=c_{1}\) and \(b x-a y=c_{2}\) are perpendicular. Assume \(a \neq 0\) and \(b \neq 0\). Text Transcription: ax + by = c_1 bx - ay = c_2 a neq 0 b neq 0

In Exercises 93-96, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. It is possible for two lines with positive slopes to be perpendicular to each other.

In Exercises 93-96, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a line contains points in both the first and third quadrants, then its slope must be positive.

In Exercises 93-96, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The equation of any line can be written in a general form.