Solved: In Exercises 7 and 8, sketch the lines through the point with the indicated

In Exercises 1-6, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 1-6, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 1-6, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 1-6, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 1-6, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 1-6, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 7 and 8, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point Slopes (3, 4) (a) 1 (b) -2 (c) \(-\frac{3}{2}\) (d) Undefined Text Transcription: -2/3

In Exercises 7 and 8, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point Slopes (-2, 5) (a) 3 (b) -3 (c) \(-\frac{1}{3}\) (d) 0 Text Transcription: -1/3

In Exercises 9-14, plot the pair of points and find the slope of the line passing through them. (3, -4), (5, 2)

In Exercises 9-14, plot the pair of points and find the slope of the line passing through them. (1, 1), (-2, 7)

In Exercises 9-14, plot the pair of points and find the slope of the line passing through them. (4, 6), (4, 1)

In Exercises 9-14, plot the pair of points and find the slope of the line passing through them. (3, -5), (5, -5)

In Exercises 9-14, 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 9-14, 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 15-18, 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

In Exercises 15-18, 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

In Exercises 15-18, 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

In Exercises 15-18, 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

Conveyor Design 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 if the vertical distance between floors is 10 feet.

Rate of Change Each of the following is the slope of a line representing daily revenue y in terms of time x in days. Use the slope to interpret any change in daily revenue for a one-day increase in time. (a) m = 800 (b) m = 250 (c) m = 0

Modeling Data The table shows the populations y (in millions) of the United States for 2000 through 2005. The variable t represents the time in years, with t = 0 corresponding to 2000. (Source: U.S. Bureau of the Census) (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.

Modeling Data The table shows the rate r (in miles per hour) that a vehicle is traveling after t seconds. (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 interval when the vehicle’s rate changed most rapidly. How did the rate change?

In Exercises 23-28, find the slope and the y-intercept (if possible) of the line. y = 4x - 3

In Exercises 23-28, find the slope and the y-intercept (if possible) of the line. -x + y = 1

In Exercises 23-28, find the slope and the y-intercept (if possible) of the line. x + 5y = 20

In Exercises 23-28, find the slope and the y-intercept (if possible) of the line. 6x - 5y = 15

In Exercises 23-28, find the slope and the y-intercept (if possible) of the line. x = 4

In Exercises 23-28, find the slope and the y-intercept (if possible) of the line. y = -1

In Exercises 29-34, find an equation of the line that passes through the point and has the indicated slope. Sketch the line. Point Slope (0, 3) \(m=\frac{3}{4}\) Text Transcription: m = 3/4

In Exercises 29-34, find an equation of the line that passes through the point and has the indicated slope. Sketch the line. Point Slope (-5, -2) m is undefined.

In Exercises 29-34, find an equation of the line that passes through the point and has the indicated slope. Sketch the line. Point Slope (0, 0) \(m=\frac{2}{3}\) Text Transcription: m = 2/3

In Exercises 29-34, find an equation of the line that passes through the point and has the indicated slope. Sketch the line. Point Slope (0, 4) m = 0

In Exercises 29-34, find an equation of the line that passes through the point and has the indicated slope. Sketch the line. Point Slope (3, -2) m = 3

In Exercises 29-34, find an equation of the line that passes through the point and has the indicated slope. Sketch the line. Point Slope (-2, 4) \(m=-\frac{3}{5}\) Text Transcription: m = - 3/5

In Exercises 35-44, find an equation of the line that passes through the points, and sketch the line. (0, 0), (4, 8)

In Exercises 35-44, find an equation of the line that passes through the points, and sketch the line. (0, 0), (-1, 5)

In Exercises 35-44, find an equation of the line that passes through the points, and sketch the line. (2, 1), (0, -3)

In Exercises 35-44, find an equation of the line that passes through the points, and sketch the line. (-2, -2), (1, 7)

In Exercises 35-44, find an equation of the line that passes through the points, and sketch the line. (2, 8), (5, 0)

In Exercises 35-44, find an equation of the line that passes through the points, and sketch the line. (-3, 6), (1, 2)

In Exercises 35-44, find an equation of the line that passes through the points, and sketch the line. (6, 3), (6, 8)

In Exercises 35-44, find an equation of the line that passes through the points, and sketch the line. (1, -2), (3, -2)

In Exercises 35-44, find an equation of the line that passes through the points, and 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 35-44, find an equation of the line that passes through the points, and 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.

Show that the line with intercepts (a, 0) and (0, b) has the following equation. \(\frac{x}{a}+\frac{y}{b}=1,\quad\ \ a\ne0,\ b\ne0\) Text Transcription: x/a + y/b = 1, a neq 0, b neq 0

In Exercises 47-50, use the result of Exercise 46 to write an equation of the line in general form. x-intercept: (2, 0) y-intercept: (0, 3)

In Exercises 47-50, use the result of Exercise 46 to write an equation of the line in general form. x-intercept: \(\left(-\frac{2}{3},\ 0\right)\) y-intercept: (0, -2) Text Transcription: (-2/3, 0)

In Exercises 47-50, use the result of Exercise 46 to write an equation of the line in general form. Point on line: (1, 2) x-intercept: (a, 0) y-intercept: (0, a) \((a \neq 0)\) Text Transcription: (a neq 0)

In Exercises 47-50, use the result of Exercise 46 to write an equation of the line in general form. Point on line: (-3, 4) x-intercept: (a, 0) y-intercept: (0, a) \((a \neq 0)\) Text Transcription: (a neq 0)

In Exercises 51-58, sketch a graph of the equation. y = -3

In Exercises 51-58, sketch a graph of the equation. x = 4

In Exercises 51-58, sketch a graph of the equation. y = -2x + 1

In Exercises 51-58, sketch a graph of the equation. \(y=\frac{1}{3} x-1\) Text Transcription: y = 1/3x - 1

In Exercises 51-58, sketch a graph of the equation. \(y-2=\frac{3}{2}(x-1)\) Text Transcription: y - 2 = 3/2 (x - 1)

In Exercises 51-58, sketch a graph of the equation. y - 1 = 3(x + 4)

In Exercises 51-58, sketch a graph of the equation. 2x - y - 3 = 0

In Exercises 51-58, sketch a graph of the equation. x + 2y + 6 = 0

Square Setting Use a graphing utility to graph the lines y = 2x - 3 and \(y=-\frac{1}{2} x+1\) in each viewing window. Compare the graphs. Do the lines appear perpendicular? Are the lines perpendicular? Explain Text Transcription: y = - 1/2 x + 1

A line is represented by the equation ax + by = 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: 5/8 y = 2/5 x + 3

In Exercises 61-66, 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 61-66, 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 61-66, 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 61-66, 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

In Exercises 61-66, 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 61-66, 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

Rate of Change In Exercises 67-70, you are given the dollar value of a product in 2008 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 2000.) 2008 Value Rate $1850 $250 increase per year

Rate of Change In Exercises 67-70, you are given the dollar value of a product in 2008 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 2000.) 2008 Value Rate $156 $4.50 increase per year

Rate of Change In Exercises 67-70, you are given the dollar value of a product in 2008 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 2000.) 2008 Value Rate $17,200 $1600 decrease per year

Rate of Change In Exercises 67-70, you are given the dollar value of a product in 2008 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 2000.) 2008 Value Rate $245,000 $5600 decrease per year

In Exercises 71 and 72, use a graphing utility to graph the parabolas and find their points of intersection. Find an equation of the line through the points of intersection and graph the line in the same viewing window. \(y=x^{2}\) \(y=4 x-x^{2}\) Text Transcription: y = x^2 y = 4x - x^2

In Exercises 71 and 72, use a graphing utility to graph the parabolas and find their points of intersection. Find an equation of the line through the points of intersection and graph the line in the same viewing window. \(y=x^{2}-4 x+3\) \(y=-x^{2}+2 x+3\) Text Transcription: y = x^2 - 4x + 3 y = -x^2 + 2x + 3

In Exercises 73 and 74, 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 73 and 74, 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 75-77, find the coordinates of the point of intersection of the given segments. Explain your reasoning. Perpendicular bisectors

In Exercises 75-77, find the coordinates of the point of intersection of the given segments. Explain your reasoning. Medians

In Exercises 75-77, find the coordinates of the point of intersection of the given segments. Explain your reasoning. Altitudes

Show that the points of intersection in Exercises 75, 76, and 77 are collinear.

Temperature Conversion 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: 0 degree C (32 degrees F) 100 degrees C (212 degrees F) 72 degrees F

Reimbursed Expenses A company reimburses its sales representatives $175 per day for lodging and meals plus \(48\text{¢}\) 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: 48 cents

Career Choice An employee has two options for positions in a large corporation. One position pays $14.50 per hour plus an additional unit rate of $0.75 per unit produced. The other pays $11.20 per hour plus a unit rate of $1.30. (a) Find linear equations for the hourly wages W in terms of x, the number of units produced per hour, for each option. (b) Use a graphing utility to graph the linear equations and find the point of intersection. (c) Interpret the meaning of the point of intersection of the graphs in part (b). How would you use this information to select the correct option if the goal were to obtain the highest hourly wage?

Straight-Line Depreciation 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, \(0 \leq x \leq 5\). (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: 0 leq x leq 5

Apartment Rental 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 if the rent is raised to $855. (c) Linear interpolation Predict the number of units occupied if the rent is lowered to $795. Verify graphically.

Modeling Data 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 y is the average test 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 test 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.

Tangent Line 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

Tangent Line 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

Distance In Exercises 87-92, 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 Ax + By + C = 0. \(\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Point: (0, 0) Line: 4x + 3y = 10 Text Transcription: (x_1, y_1) Distance = |Ax_1 + By_1 + C| / sqrt A^2 + B^2

Distance In Exercises 87-92, 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 Ax + By + C = 0. \(\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Point: (2, 3) Line: 4x + 3y = 10 Text Transcription: (x_1, y_1) Distance = |Ax_1 + By_1 + C| / sqrt A^2 + B^2

Distance In Exercises 87-92, 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 Ax + By + C = 0. \(\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Point: (-2, 1) Line: x - y - 2 = 0 Text Transcription: (x_1, y_1) Distance = |Ax_1 + By_1 + C| / sqrt A^2 + B^2

Distance In Exercises 87-92, 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 Ax + By + C = 0. \(\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Point: (6, 2) Line: x = -1 Text Transcription: (x_1, y_1) Distance = |Ax_1 + By_1 + C| / sqrt A^2 + B^2

Distance In Exercises 87-92, 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 Ax + By + C = 0. \(\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Line: x + y = 1 Line: x + y = 5 Text Transcription: (x_1, y_1) Distance = |Ax_1 + By_1 + C| / sqrt A^2 + B^2

Distance In Exercises 87-92, 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 Ax + By + C = 0. \(\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Line: 3x - 4y = 1 Line: 3x - 4y = 10 Text Transcription: (x_1, y_1) Distance = |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 Ax + By + C = 0 is \(\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) Text Transcription: (x_1, y_1) Distance = |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.

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.

True or False? In Exercises 99 and 100, 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

True or False? In Exercises 99 and 100, 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.